On structures of certain -we11-posed mixed problems
for hyperbolic systems of first order
L^{2}
By Toshio OHKUBO and Taira SHIROTA
\S 1. Introduction and results.
Let P be an -strictly hyperbolic 2m\cross 2m -system of differential opera-cylinder
. Let B be an
tors of first order defined over a
m\cross 2m -system of functions defined on the boundary
. We
of
consider the following mixed problem:
x_{0}
C^{\infty}
R^{1}\cross\Omega\subset R^{n+1}
R^{1}\cross\Omega
\Gamma
u(x)=f(x)
B(x)u(x)=g(x)
u(x)=h(x)
p\{x, D)
(P, B)
where
x=(x_{0}, x_{1}, \cdots, .x_{n})
,
x\in R^{1}\cross\Omega
(x_{0}>0)
,
x\in\Gamma
(x_{0}>0)
,
x\in R^{1}\cross\Omega
(x_{0}=O)
,
D_{j}=-^{1} \frac{\partial}{\partial x_{j}}=\frac{1}{i}\frac{\partial}{\partial x_{j}}\sqrt{-1}
and
D=(D_{0}, \cdots, D_{n})
. We
consider -well-posedness of (P, B) in the follwing sense:
DEFINITION 1. 1. The problem (P, B) is said to be -well-posed if there
exist positive constants C, T such that for every f\in H_{1}((-\infty, T)\cross\Omega) with
f=0(x_{0}<0) , g=0 and h=O , there exists a unique solution u\in H_{1}((-\infty, T)
with u=O(x_{0}<0) which satisfifies the inequality:
L^{2}
L^{2}
\cross\Omega)
\int_{0}^{T}||u||_{0}^{2}
, \Omega dt\leqq C\int_{0}^{T}||f||_{0,12}^{2}
dt,
where H_{k}(G) is the Sobolev space with its norm . , G .
The aim of the present article is, under somewhat strong but general
restriction on the operator P, to describe, in terms of the cotangent space
, the relations among the coefficients of boundary operator
of
problem (P, B) . These relations are useful for the
B for
investigation of the propagation of singularities of solutions for our pr0blems. For example they determine whether there exist lateral waves or
not. Applying the relations we prove the existance of the solution of our
problem. But in contrast with the recent development of the Cauchy pr0blems, we must essentially use the energy estimate, because of the existance
of glancing rays with the associated non-vanishing reflection coefficients in
these cases
||
R^{1}\cross\partial\Omega=\Gamma
L^{2}- we11-\overline{p}osed
||_{k}
On structures of certain
Put
L^{2}
-well-posed mixed problems for hyperbolic systems
83
respectively and let
be the prinx=(x_{0},
x’,
x_{n}) .
cipal part of P (x,
), where
By
is a covector of
virture of localization and certain coordinate transformations, we may restrict ourselves to the case where
C_{\pm}=\{\tau\in C^{1} ; {\rm Im}<>0\}
(\tau, \sigma, \lambda)
\lambda
\tau,
\Omega
P^{0}(x, \tau, \sigma, \lambda)
\sigma,
,
=R_{+}^{n}=\{(x’, x_{n});x_{n}>0 x’\in R^{n-1}\}
R^{1}\cross\Omega
=R_{+}^{n+1}=\{(x_{0}, x’, x_{n});x_{n}>0
\Gamma=R^{n}=\{x’\in R^{n+1} ; x_{n}’=O\}
,
,
n\geqq 2
,
(x_{0}, x’)\in R^{n}\}
,
.
Then our assumptions are the following:
(I)
The coefficients of
and B are real, belong to
and are constant outside some compact set of
.
multiplicity
The
of the real roots
of the characteristic
equation
is at most double and there is at most one
real double root, for every
.
(P, B) is normal, that is, (i)
for every (x’ ,
,
, (ii) rank B(x’)=m for every x’\in I
(II) Let
be the Lopatinskii determinant of (P^{0}, B) (for definition, see \S 4.)
If
for a point
such that there
are no real double roots of
, it holds for small \gamma>0
that
P^{0}
\alpha)
C^{\infty}(R^{1}\cross\overline{\Omega})
R^{1}\cross\overline{\Omega}
\beta)
\lambda(xd, \tau, \sigma)
\det P^{0}(x’, \tau, \sigma, \lambda)=O
(x’, \tau, \sigma)\in\Gamma\cross(R^{n}\backslash \langle O\})
\det P^{0}(x’, O, \sigma, \lambda)\neq O
\gamma)
\sigma
\lambda)\in I
\cross
(R^{n}\backslash \{O\})
R(x’, \tau, \sigma)
R(x^{0}, \tau^{0}, \sigma^{0})=O
\alpha)
(x^{0}, \tau^{0}, \sigma^{0})\in I
\lambda
\cross(R^{n}\backslash \{O\})
\det P^{0}(x^{0}, \tau^{0}, \sigma^{0}, \lambda)=O
|R(x^{0}, \tau^{0}-i\gamma, \sigma^{0})|\geq C^{\gamma}-
with some constant
one real simple root
some neighborhood
If
is a real double root
. Furthermore if there is at least
, R(x’, \tau, \sigma) vanishes only for {\rm Im}\tau=O in
C=C(x^{0}, \tau^{0}, \sigma^{0})>O
\lambda(xd, \tau^{0}, \sigma^{0})
U(x^{0}, \tau^{0}, \sigma^{0})\subset\Gamma\cross C^{1}\cross R^{n-1}
R(x^{0}, \tau^{0}, \sigma^{0})=O
\beta)
\lambda
of
for a point
.
(x^{0}, \tau^{0}, \sigma^{0})\in I
\det P^{0}(x^{0}, \tau^{0}, \sigma^{0}, \lambda)=O
\cross(R^{n}\backslash \{O\})
, for small
such that there
\gamma>0
\frac{1}{2}
|R(x^{0}, \tau^{0}-i\gamma, \sigma^{0})|\geq C
with some constant
. Furthermore if there is at least
eal
simple root , the rank of
one r
at the zeros of
in some neighborhood
is equal to
co\dim of \{(x’, \sigma);R(x’, \theta(xd, \sigma), \sigma)=O\} in R^{2n-1} ,
where
denotes a real valued function such that
has only a real double root on the surface
with
(see Corollary 3. 1.) Here the zero set \{(x’, \sigma);R(x’, \theta(xd, \sigma), \sigma)=O\} in some
is preassumed to be a regular submanifold of R^{2n- 1} .
C=C(x^{0}, \tau^{0}, \sigma^{0})>0
\lambda
He” ss(x\sigma)R(x’, \theta(xd, \sigma), \sigma)
R(x’, \theta(x’ \sigma), \sigma)
U(x^{0}, \tau^{0}, \sigma^{0})
\theta(xd, \sigma) \in C^{\infty}
P^{0}(x’, \tau, \sigma, \lambda)=O
\lambda
U(x^{0}, \tau^{0}, \sigma^{0})
\tau=\theta(xd, \sigma)
\tau^{0}=\theta(xd, \sigma^{0})
T. Ohkubo and T. Shirota
84
satisfying the first condition of ), if there
In the point
, then the reflection
is at least one non-real root of
whenever ,
is real in some
coefficient
II
and reflection coefare real and R(x’, \tau, \sigma)\neq O (for definitions of
ficients see \S 3 and \S 4, respectively).
(Ill) Any constant coefficients problems (P, B)_{x’} resulting from freezing
the coefficients at boundary points x’ are -well-posed.
The main result in this article is the following
MAIN THEOREM. Assume the conditions (/), (II) and (III), then the
variable coeffiffifficients problem (P, B) is also -well-posed.
To prove Main theorem, we use the following
THEOREM 1. 1. Under the conditions (/), (II) and (III) the following
, r_{k}>0 :
a priori estimate holds with some constants
(x^{0}, \tau^{0}, \sigma^{0})
\gamma)
\beta
\det P^{0}(x^{0}, \tau^{0}, \sigma^{0}, \lambda)=O
\lambda
b_{II}^{-}
U(x^{0}, \tau^{0}, \sigma^{\rho}’)
(x’, \tau, \sigma)
\tau
\lambda_{II}^{\pm}(x’, \tau, \sigma)
\lambda_{II}^{\pm}(x’, \tau, \sigma)
L^{2}
L^{2}
C_{k}
(1. 1)
||
for every
defifined
\gamma\geq\gamma_{k}
,
Pu
||_{k,T}+|Bu|_{k+\frac{1}{2}}
u\in H_{k,\gamma}(R_{+}^{n+1})
,r
\geq C_{k}\gamma||u||_{k,\gamma}
and integer
k\geq O
{the norms used here are
in \S 2.)
Our method deriving Main theorem is applicable to the case where the
boundary operator B(x’) is complex. Main theorem is also valid if we
and ) instead of the conditions (II)
assume the following conditions (II)
and ):
such that
for a point
(II)
If
, for small \gamma>0
there is a double root of
\mathcal{T}’
\beta’)
\beta)
\gamma
\beta’)
(x^{0}, \tau^{\mathfrak{y}}, \sigma^{0})\in I
R(x^{0}, \tau^{0}, \sigma^{0})=O
\cross(R^{n}\backslash \{O\})
\det P^{0}(x^{0}, \tau^{0}, \sigma^{0}, \lambda)=O
\lambda
|R(x^{0}, \tau^{0}-i\gamma, \sigma^{()})|\geq C\gamma^{\frac{1}{2}}
. Let us consider R as a funcwith some come constant
. Then by the implicit function theorem we can
tion of
decompose R as
C=C(x^{9}, \tau^{0}, \sigma^{0})>0
(x’, \sqrt{\tau-\theta(x’,\sigma)}, \sigma)
R(x’, \tau, \sigma)=r(x’,
where \sqrt{1}=-1 and
for some constant
,
\sqrt{\tau-\theta(x’,\sigma)}
r(x^{0}, O, \sigma^{0})\neq 0
and
\sigma
)
(\sqrt{\tau-\theta(x’,\sigma)}-D(x’, \sigma)
D(x^{0}, \sigma^{0})=0.
)
Now we assume that
C=C(x^{0}, \tau^{0}, \sigma^{0})>0
{\rm Re} D(x’, \sigma)\geq C({\rm Im} D(x’, \sigma)
(1. 2)
)
2
in the case (a),
or
-{\rm Im} D(x’, \sigma)\geq C
in some neighborhood
ma 3. 1 respectively.
(Re
U(x^{0})\cross U(\sigma^{0})
D(x’, \sigma))^{2}
, according to
in the case (b)
the case (a) or (b) in Lem-
On structures of certain
L^{2}
-well-posed mixed problems
for hyperbolic
85
systems
In the point satisfying the first condition of ), if there is at
least one real simple root , the zero set of {\rm Re} D(x’, \sigma) (or {\rm Im} D ( x’ , )) in
some
is preassumed to be a regular submanifold of R^{2n-1} and the
(or {\rm Im} D ( x’ , )) at the zeros of {\rm Re} D(x’, \sigma) (or
rank of
(II)
\mathcal{T}’)
\beta’
\lambda
\sigma
U(x^{0}, \sigma^{0})
\sigma
He” ss(x\sigma){\rm Re} D(x’,, \sigma)
is equal to
{\rm Im} D(x’, \sigma))
co\dim
of
or
{
(x’, \sigma)
;
{\rm Re} D(x’, \sigma)=O\}
\{(x’, \sigma)
;
{\rm Im} D(x’, \sigma)=O\}
in
R^{2n-1}
in the case (a) or (&), respectively.
Here we remark that the conditions (II) and ) imply the conditions
(II)
and ) when we are restricted to the real case (see subsection 10.2),
and that the last condition of (II)
and (II)
are omitted, if (6. 5) and
(10. 6) are satisfied, respectively. It should also be noted that the conditions
(I), (II) and (III) are invariant for certain coordinate transformations, hence
Main theorem is applicable to problems defined on any smooth
.
Throughout this article we assume the condition (I). Section 2 and
section 3 are a summary of elementary facts which are used in later sections. In section 4, a necessary and sufficient condition for -well-posedness
of the constant coefficients problem is shown in terms of ‘coupling’ coefficients. Using it we investigate, in section 5 and 6, the structure of
variable coefficients problem under the condition (III). Section 7 is devoted
to the estimates of the problems for
first order systems. The ideas
used there play an essential role in the proof of Theorem 1. 1 which is
shown in section 8. Section 9 is concerned with a dual problem to (P, B) .
In subsection 10. 1 an example showing the necessity of the condition (II)
is presented, and in subsection 10. 2 the generalization to the case where
the boundary operator B(x’) is complex valued is considered.
Second author lectured the outline about this theme ([12]) and first
author gave the proof in detail. Authors are indebted to Dr. R. Agemi
for a number of helpful discussion of these problems and for revises of
the manuscript.
\beta)
\beta’)
\gamma
\gamma’
\gamma’)
\beta)
R^{1}\cross\Omega
L^{2}
2\cross 2
\beta)
\S 2. Preliminaries.
2. 1. The following spaces are used in this paper. ([6], [8]) Let us
define for
and real numbers p , q the Hilbert spaces of functions:
\mathcal{T}\neq 0
H_{p,q;f}(R^{n+1})=\{u\in\sigma\Delta’(R_{n+1})
H_{q,\}}(Il^{n})=\{v\in c\overline{\Lambda}’(R^{n})
;
;
e^{-\gamma x_{0}}u\in H_{p}
, Q(R_{n+1})\}
e^{-\gamma x_{0}}v\in H_{q}(R^{n})\}
,
T. Ohkubo and T. Shirota
86
with their norms defined by
||u||_{p}
, q;
\gamma,R^{n+1}=\int_{R^{n+1}}(\mathcal{T}^{2}+|\xi|^{2})^{p}(\mathcal{T}^{2}+|\xi’|^{2})^{q}|e^{\hat{-\gamma x_{0}}}u(\xi)|^{2}d\xi
|v|_{q}
,r
,
= \int_{R^{n}}(\gamma 2+|\xi’|^{2})^{q}|\hat{e^{-\gamma x_{0}}}v(\xi’)|9.d\xi’
, .,
and their inner products ,
, respectively, where H_{p,q}(R^{n+1}) ,
H_{q}(R^{n}) are Sobolev spaces, \xi=
is a covector of x=(x’ ,
, a nd
,
are the Fourier transforms
H_{p.q;\gamma}(Rn+1+)
respectively. Moreover, let
u,
be the set of all
of
such that there exist distributions U\in H_{p,q;f}(R^{n+1}) with U=u
. The norm of u is defined by
in
(\cdot
\cdot)_{p,q;\gamma,R^{n+1}}
\langle
\cdot\rangle_{q}
\gamma
(\xi’, \xi_{n})=
\hat{e^{-\gamma x_{0}}}u(\xi)
x_{n})=(x_{0}, x’, x_{n})\in R^{1}\cross\Omega
(\tau, \sigma, \lambda)\in R^{n+1}
e^{\hat{-\gamma x_{0}}}\grave{v}(\xi’)
e^{-*x_{0}}v
e^{-\gamma x_{0}}
u\in 6\overline{\Lambda}’(R_{+}^{n+1})
R_{+}^{n+1}
||u||_{p,q}j\gamma,R_{+}^{n+1}
=
||
\inf_{U}
U||_{p}
From now on, for simplicity we denote by
and by ,
by .
the norm .
Note that
||
||
||_{p,\gamma}
||_{p,0;r},R_{+}^{n+1}
H_{p,\gamma}(R_{+}^{n+1})
and
||u||_{k}
,f
k\geq 0
=\{u
;
,
the space
the inner product respectively.
H_{p,0;r}(R_{+}^{n+1})
H_{p,\gamma}(R_{+}^{n+1})
\cdot)_{p,\gamma}
e^{-\gamma x_{0}}u\in H_{p}(R_{+}^{n+1})\}
is equivalent to
(
if
(\cdot
, q ; r ,Rn+1
\sum_{j+l=k}\int_{0}^{\infty}|D_{n}^{j}u(\cdot, x_{n})|_{l,f}^{2}dx_{n})-
is an integer.
2. 2. We consider the following pseud0-differential operator with pa. Let
be
rameters \gamma>0 and
.
a function which is independent of x outside some compact set of
( k : real) means that for every ,
there exists
That
such that
a constant
x_{n}
a(x’, x_{n}, \xi’, \gamma)\in C^{\infty}(\overline{R^{n+1}}\cross((R^{n}\cross\overline{R_{+}^{1}})\backslash \{O\})
\overline{R_{+}^{n+1}}
a(x’, x_{n}, \xi’, \gamma)\in S_{+}^{k}
\beta
\alpha
C_{a,\beta}>O
|D_{x}^{\alpha}
, D_{\xi}^{\beta},a(x’, x_{n}, \xi’, \gamma)|\leq C_{\alpha,\beta}(\gamma^{2}+|\xi’|^{2})^{\frac{1}{2}(k-|\beta|)}
. Matrix functions in
are defined
for every
called a symbol we associate
similarly. With a matrix function
,
a pseud0-differential operator A (x, D’, ) defined for any vector
, by the formula
which means that the components of v belong to
(x, \xi’, \gamma)\in R_{+}^{n+1}\cross R^{n}\cross R_{+}^{1}
S_{+}^{k}
A\in S_{+}^{k}
v\in H_{s,\gamma}(R^{n})
\gamma
H_{s},\cdot(R^{n})
A(x, D’, \gamma)v(x’)
=
where
\tau=\eta-i\gamma
and
(2 \pi)^{-n}e^{\gamma x_{0}}\int R^{n}e^{ir’ x_{0}+i\sigma x’}A(x, \eta, \sigma, \gamma)\hat{v}(\tau, \sigma)d\eta
\xi’=(\eta, \sigma)
. Note that if
da
On strvatures of certai n
L^{2}
-well-posed mixed problems for hyperbolic systems
A(x, \eta, \sigma, \gamma)=A
then it holds that if A (x,
\tau,
\sigma
( x,
r,
\eta-i
\sigma
)
87
=A(x, \tau, \sigma)
) is analytic with respect to
\tau
,
A(x, D’, \gamma)v(x’)=A(x, D’)v(x’)
where
and A(x, D’) is a usual pseud0-differential operator.
The basic properties of usual pseud0-differential operators hold anal0gously for this case :
Lemma 2. 1. ([9], [5])
For every real s there exist positive constants and C_{s}ind\varphi endmt
it holds that
of v such that for every
and
(i) for real k,
v\in H_{s,\gamma}(R^{n})
\alpha)
\gamma_{s}
\gamma\geq\gamma_{s}
v\in H_{s+k,\gamma}(R^{n})
A\in S_{+}^{k}
|A(x, D’, \gamma)v|_{s,\gamma}\leq C_{s}|v|_{s+k,f}
(ii)
adjoint
for real
of
A\in S_{+}^{k}
for
(
,
A^{*}(x, \xi’, \gamma)
with respect to
A(x, D’, \gamma)
|
(iii)
k,
\langle
.,
A^{*}(x, D’, \gamma)-A^{\#}(x, D’, \gamma)
A_{i}\in S_{+}^{k_{t}}
,
k_{i}
,
the adjoint matrix,
, r and
\cdot
)
: real, i=1,2,
A^{\#}
the fo real
v\in H_{s+k,\gamma}(R^{n})
\rangle_{0}
v|_{s+1,\gamma}\leq C_{s}|v|_{s+k}
,
\gamma,\cdot
A_{3}(x, \xi’, \gamma)=A_{1}A_{2}\in S_{+}^{k_{1}+k_{2}}
and
v\in
H_{s+k_{1}+k_{2},\gamma}(R^{n})
|
(
A_{3}(x, D’, \gamma)-A_{1}(x, D’, \gamma)\cdot A_{2}(x, D’, \gamma)
\leq C_{s}|v|_{s+k_{1}+k_{2}}
(iv)
Let
,
and
\gamma\geq\gamma_{0}
{\rm Im}
v|_{s+1,\gamma}
r^{\iota}
A(x, \xi’, \gamma)=A^{*}(x, \xi’, \gamma)\in S_{+}^{k}
such that for every
)
, k : real.
Then there exist C,
\gamma_{0}>0
v\in H_{k,\gamma}(R^{n})
\langle A(x, D’, \gamma)v, v\rangle_{0,f}\leq C|v|_{k-1}^{2}
.
\overline{z}’\gamma
\beta)
exist C,
Let
and A(x, \xi’, \gamma)=A^{*}(x, \xi’, \gamma)\geq 0
such that for
for
A\in S_{+}^{0}
\gamma_{1}>O
\gamma\geq\gamma_{0}>O
. Then there
\gamma\geq\gamma_{1}
{\rm Re}
\langle A(x, D’, \gamma)v, v\rangle_{0,\gamma}\geq-C|v|_{-\frac{1}{2},r}^{2}
.
This is a Garding s sharp form . From this the following hold.
(i) Let
) =A^{*}\geq C with C>0 . Thm there exists
and A (x,
a
such that for
A\in S_{+}^{J}|
\xi’,
\mathcal{T}_{1}>0
\gamma
\gamma\geq\gamma_{1}
{\rm Re}
(ii) Let
real s there exist
and
A\in S_{+}^{\theta}
C_{s}
,
.
with C>O .
\langle A(x, D’, \gamma)v, v\rangle_{I),f}\geq 2^{-1}C|v|_{0,r}^{2}
|\det A(x, \xi’, \gamma)|\geq C
T_{s}>0
such that for
\gamma\geq\gamma_{s}
|A(x, D’, \gamma)v|_{s,\gamma}\geq C_{s}|v|s,\gamma
.
Thm
for every
T. Ohkubo and T. Shirota
88
Let A (x,
(iii)
\xi’,
\gamma
)
and
\in S_{+}^{0}
A^{-1}
such that for
\gamma)^{-1}=A^{-1}(x, D’, \gamma)+T in
which
there exist
C_{s}
,
\gamma_{s}>0
(x,
\xi’,
)
. Then for every real s
(x, D’, ) has an inverse A (x, D’ ,
\in S_{+}^{0}
\gamma
\gamma\geq\gamma_{s}A
\gamma
satisfifies
H_{s,\gamma}
|A^{-1}x|_{s,\gamma}\leq C_{s}|v|_{s}
|
for every
v\in H_{s,\gamma}(R^{n})
Let
Then there exist
(iv)
A\in S_{+}^{1}
C_{1}
,
Let
C_{0}
,
\gamma_{c}>0
,
there exist
A\in S_{+}^{1}
and
,
\gamma_{1}>0
)
(vi) Let a (x,
, where
\xi’,
\gamma
, where
and
C_{0}
,
\mathcal{T}_{0}>0
.
\gamma\geq\gamma_{1}
.
, where for sme
for every
A_{2}(x, \xi’, \gamma)=A_{2}^{*}
\gamma\geq\gamma_{0}
. Then
\gamma\geq\gamma_{1}
\langle A(x, D’, \gamma)v, v\rangle_{0,f}\geq c_{1}r
|v|_{0}^{2}
,r
.
be a scalar fu nction such that
. Then there exists a constant
\in S_{+}^{Q}
\epsilon>0
\gamma\geq\gamma_{0}>O
\gamma\geq\gamma_{0}
A(x, \xi’, \gamma)=A_{1}(x, \xi’, \gamma)+iA_{2}(x, \xi’, \gamma)
such that for
{\rm Re}
for
such that for
for
\langle A(x, D’, \gamma)v, v\rangle_{0,r}\geq C_{1}\gamma|v|_{0,f}^{2}
and
\gamma_{1}>0
\gamma
A(x, \xi’, \gamma)=A^{*}\geq C_{0}\gamma
A_{1}(x, \xi’, \gamma)=A_{1}^{*}\geq C_{0}\gamma
C_{1}
,
,
.
{\rm Re}
(v)
Tv|_{s,\gamma}\leq C_{s}|v|_{s-1}
,f
|A(x, \xi’, \gamma)|\leq C_{0}-\epsilon
\gamma_{1}>0
such that for
\gamma\geq\gamma_{1}
|a(x, D’, \gamma)
v|_{0,\gamma}\leq C_{0}|v|_{0,\gamma}
.
In the present paper, our process of proofs of Main theorem are
rging’s sharp
carried somewhat long and classically, in order to use the
form directly and decompose the original problem into boundary value
and R^{n} in which we are interested.
problems well defined over
Furthermore for the sake of simplicity of description we denote by
) and by operators a (x, D’ ) on H_{r}., the
) the symbols a (x,
a (x,
corresponding pseud0-differential operators a(x, D’, \gamma) respectivevely.
2. 3. We reduce the global estimate (1. 1) into the micr0-local one as
usual ([5]). Since lower order terms of the operator P do not affect the
validity of the estimate (1. 1), we have only to consider the boundary value
problem :
G[mathring]_{a}
R_{+}^{n+1}
\gamma
\eta-i\gamma,
\eta,
\sigma
\sigma,
P^{0}(x, D)u(x)=(ED_{n}-A(x, D’))u=f,
in
R_{+}^{n+1}
B(x’)u(x’, 0)=q(x’) ,
in
R^{n}
,
(P^{0}, B)\{
) = A( x\}
A A(x} and |.E is
where the symbol of A(x, D’) is A (x,
the 2m\cross 2m identity matrix.
be the compact set outside which A_{i}(x)i=O , ,
First, let
n- 1 and B(x’) are constant. Let us take a finite number of points
\tau,
K\subset R^{n+1}
\sigma
\tau+\sum_{i=1}^{n-1}
\sigma_{i}
\cdots
On structures
of certain
L^{2}
such that the spheres
, where
j=1, , j_{0}-1 are a covering of
Take a finite partition of unity
such that
, .,
\{x^{j}\}_{j=1}
,
\cdots
89
-well-posed mixed problems for hyperbolic systems
K\cap\Gamma
\cdots
;
},
is sufficiently small.
S_{2\epsilon_{0}}(x^{j})=\langle x\in R^{n+1}
, j_{0}-1\subset\Gamma
\epsilon_{0}>O
|x-xf|<2\epsilon_{0}
\{\phi^{j}\}_{j=0}
j_{0}
\cdot
supp\phi^{j}\subset S_{2\cdot 0}
j=1 ,
(x^{j})
, j_{0}-1
\cdots
,
j_{0}
on
\sum\phi^{f}(x)=1
\overline{R_{+}^{n+1},}
f=0
\phi^{j}\in C^{\infty}(R_{+}^{n+1})
j=O ,
1, for
0, for
\cdots
,
x_{n}\geq_{E}^{g}\epsilon_{0}
\phi^{j_{0}}(x)=\phi^{j_{0}}(x_{n})=\{
x_{n}\leq\epsilon_{0}
for
\phi^{0}(x)=O
A_{i}(x)
, i=O ,
\cdots
n-1
,
,
,
x_{n}\geq_{Z^{6}0}^{3}
,
,
j_{0}
and B(x’) are constant in
supp\phi^{0}(x)
.
(x,
we construct the corresponding operators B^{j}(x’) and
=A_{0}^{j}(x)\tau+
D)=ED_{n}-A^{j}(x, D’) where the symbol of
)
(x,
(x, D’ ) i s
n-l
A ji(x) , which have the following properties:
For
P^{f}
1\leq j\leq j_{0}-1
A^{j}
A^{f}
\tau,
\sigma
\sigma_{i}
\sum_{\dot{\iota}=1}
(2. 1)
(i)
A^{j}
(ii)
A^{j}
(iii)
A^{j}
(x,
(x,
(x,
\tau,
\sigma
\tau,
\sigma
\tau,
\sigma
and
)
\in S_{+}^{1}
)
=A (x,
B^{j}(x’)\in S_{+}^{0},\cdot
) and
B^{j}(x’)=B(x’)
) =A(x^{j}, \tau, \sigma) and
B^{j}(x’)=B(x^{j})
\tau,
\sigma
for every
for every
x\in S_{2\cdot 0}(x^{j})
.
x\not\in
,
S_{\Sigma^{5}0},.(x^{j})
.
,
is done as follows: Choose a function
The construction of
, O<\beta_{1}(t)<1 for 1<
such that equals 1 for t\leq 1, O for
P^{j}
B^{j}
\beta_{1}(t)\in C^{\infty}(\overline{R_{+}^{1}})
t< \frac{5}{4}
. For
x\in\overline{R_{+}^{n+1}}
t \geq\frac{5}{4}
put
\overline{x}^{j}(x)=x^{j}
\dagger
\beta_{1}(|x-x^{j}|/2\epsilon_{0})(x-x^{j})
and let
A^{f}
(x,
\tau,
\sigma
) =A(\tilde{x}^{j}(x), ,
\tau
\sigma)
),
B^{j}(x’)=B(\overline{x^{j}}(x’))
Then it is seen that these operators satisfy the required properties (2. 1).
Now we have the following
LEMMA 2. 2. Suppose the condition (III) and that there are constants,
the inequalities
and
C, T_{0}>0 such that for every
\gamma\geq\gamma_{0}
(2. 2)
||Pj\phi^{j}u||_{0,\gamma}+|B^{j}\phi^{j}u
.
|_{z}1
,r
u\in H_{1,\gamma}(R_{+}^{n+1})
\geq C\gamma ||\phi^{j}u||_{0}
,f ’
j=1,
\cdots
, j_{0}-1 ,
hold. Then the estimate (1. 1) holds for k=O.
PROOF. The strict hyperbolicity of P^{J}(x, D) implies that there exist
and
constants , C>0 such that for every
\gamma_{\bigcap_{1}}
\gamma\geq\gamma_{0}
u\in H_{1,\gamma}(R_{+}^{n+1})
T. Ohkubo and T. Shirota
90
(2. 2. 1)
||P0(x, D)\phi^{j_{0}}u||_{0,r}\geq C\gamma||\phi^{f_{0}}u||_{0,f}
From property (2. 1) (ii), it follows that for j=1,
P^{j}(\phi^{j}u)=P^{0}(\phi^{j}u)=\phi^{j}P^{0}u+[P^{0}, \phi^{j}]
\cdots
.
, j_{0}-1
u
B^{j}(\phi^{j}u)=B(\phi^{j}u)=\phi^{j}Bu
in
R_{+}^{n+1}
on
\Gamma
.
.
Furthermore from the condition (III), (1. 1) holds for constant coefficients
problems. Hence it follows from (2. 2) and (2. 2. 1) that
C\gamma||\phi^{j}u||_{0,f}\leq||\phi^{j}P^{0}u+[P0, \phi^{j}] u||_{0,\tau}+|\phi^{j}Bu|_{z}1
\leq||P^{0}u||_{0,\gamma}+|Bu|_{\frac{1}{2}}
for j=O ,
\cdots
,
j_{0}
.
,r
,r
+||u||_{0,\gamma}
Summing up these, we have
C \gamma||u||_{0,r}=C\gamma||\sum_{j=0}^{j_{0}}\phi^{j}u||_{0,\gamma}
\leq C’(||P^{0}u||_{0,\gamma}+|Bu|_{\frac{1}{2},r}+||u||_{0},f)
is taken sufficiently large we obtain (1. 1).
Secondly, let
closure. Let us take a finite number of points
,
for
for
, where
is a covering of
-functions on -such that
be
Let
If
1
\gamma
and
\Sigma_{-}=\{\tau’, \sigma’)\in C_{-}\cross R^{n-1} ; |\tau’|^{2}+|\sigma’|^{2}= 1\}
\{(\tau_{k}’, \sigma_{k}’)\}_{k=1}
k\leq k_{0}
{\rm Im}\tau_{k}=O
\overline{\Sigma_{-}}
supp\psi_{k}’\subset S_{2*_{0}}(\tau_{k}’, \sigma_{k}’)
k_{1}\subset\overline{\Sigma}
\{S_{2\cdot 0}(\tau_{k}’, \sigma_{k}’)\}_{k=1}
,...,
k_{1}
.
on
\overline{\Sigma_{-}}
. Put
\psi_{k}(\tau, \sigma)=\psi_{k}’(\Lambda_{\overline{r}}^{1}\tau, \Lambda_{f}^{-1}\sigma)
where
..,
\overline{\Sigma}
\sum_{k=1}^{k_{1}}(\psi_{k}’(\tau’, \sigma’))^{2}=1
and
and
\cdot
S.(\tau_{k}’, \sigma_{k}’)= \{(\tau’, \sigma’)\in\overline{\Sigma_{-}};|\tau’-\tau_{k}’|^{2}+|\sigma’-\sigma k’|^{2}<\epsilon^{2}\}
C^{\infty}
\{\psi_{k}’(\tau’, \sigma’)\}
k_{0}+1\leq k\leq k_{1}
{\rm Im}\tau_{k}’\leq-3\epsilon_{0}
,
be its
-such that
\overline{\Sigma_{-}}
\Lambda_{\gamma}^{2}=|\tau|^{2}+|\sigma|^{2}=\gamma^{2}+|\xi’|^{2}
for
(\tau, \sigma)\in\overline{C_{-}}\cross R^{n-1}
. Then we have
\psi_{k}\in S_{+}^{J}
.
For each j=1, , j_{0}-1, it is possible to construct the corresponding
(x, D) =ED_{n}-A_{k}^{j}(x, D’) for k=1,
operators
, , such that the symbols
(x, D’ ) fulfill the following properties:
) of
(x,
\cdots
P_{k}^{j}
A_{k}^{j}
\tau,
(2. 3)
k_{1}
A_{k}^{j}
\sigma
(i)
(ii)
\cdots
A_{k}^{j}(x, \tau, \sigma)\in S_{+}^{1}
,
A_{k}^{j}(x, \tau, \sigma)=A^{f}(x, \tau, \sigma)
hence
A_{k}^{f}(x, \tau, \sigma)=A(x, \tau, \sigma)
The way of construction of
Case 1. k\geq k_{0}+1 .
put
For
(\tau’, \sigma’)\in\overline{\Sigma_{-}}
for
for
A_{k}^{j}
(x,
(x, \tau\Lambda_{f}^{1}, \sigma\Lambda_{\gamma}^{1})\in R_{+}^{n+1}\cross S_{2\text{\’{e}}_{0}}(\tau_{k}’, \sigma_{k}’)
(x, \tau\Lambda_{\gamma}1, \sigma\Lambda_{\gamma}^{1})\in
\tau,
\sigma
) is as
S2.0
, and
(x^{j})\cross S_{2*_{0}}(\tau_{k}’, \sigma_{k}’)
follows:
.
On structures of certain
(2. 4)
L^{2}
-well-posed mixed problems for hyperbolic systems
\tilde{\tau}_{k}(\tau’, \sigma’)=[\tau_{k}’+\beta_{1}((|\tau’-\tau_{k}’|^{2}+|\sigma’-\sigma k’|^{2})^{-}/2\epsilon_{0}
)
91
(\tau’-\tau_{k}’)]\cdot\delta^{-1}
\tilde{\sigma}_{k}(\tau’, \sigma’)=[\sigma_{k}’+\beta_{1}((|\tau’-\tau_{k}’|^{2}+|\sigma’-\sigma_{k}’|^{2})^{\frac{1}{2}}/2\epsilon_{0})(\sigma’-\sigma_{k}’)]\cdot.\delta^{-1}
where
(2. 5)
for
of
(
2.
is chosen such that
\delta(=\delta(\tau’, \sigma’))
A_{k}^{j}(x, \tau, \sigma)=A^{f}(x,\tilde{\tau}_{k}(\tau\Lambda_{f}^{-1}, \sigma\Lambda_{f}^{1})
,
\tilde{\sigma}_{k}(\tau\Lambda_{f}^{-1}, \sigma\Lambda_{\overline{r}}^{1}))
.
\Lambda_{r}
. The n
fulfill (2. 3), by virture of the homogeneity
. Remark that
of order 1 in
(\tau, \sigma)\in\overline{C_{-}}\cross Jt^{n-1}
A^{j}
. Put
(\tilde{\tau}_{k},\tilde{\sigma}_{k})\in\overline{\Sigma_{-}}
A_{k}^{f}
(\tau, \sigma)
(i)
4)’
mapps
(\tilde{\tau}_{k},\tilde{\sigma}_{k})
\Sigma
-into
S_{\frac{\epsilon}{2}}.(\tau_{k}’, \sigma_{k}’)0
(ii)
S_{\frac{5}{2}0}.(\tau_{k}’, \sigma_{k}’)\subset\{(\tau’, \sigma’)\in\Sigma_{-};
(iii)
(\tilde{\tau}_{k},\tilde{\sigma}_{k})=
Case 2.
Let
t\geq 2 and
k\leq k_{0}
{\rm Im}\tau’\leq-_{Z}^{1}\epsilon_{0}\}
for every
(\tau’, \sigma’)
’
,
(\tau’, \sigma’)\in S_{2\cdot 0}(\tau_{k}’, \sigma_{k}’)
.
.
be a function such that equals to
for 1<t<2 . For
\beta_{2}(t)\in C^{\infty}(\overline{R_{+}^{1}})
1\leq\beta_{2}(t)\leq 2
t
for
t\leq 1,1
for
-put
(\tau’, \sigma’)=(\eta’-i\mathcal{T}’, \sigma’)\in\overline{\Sigma}
(2. 6)
\rho_{k}=(|\eta’-\tau_{k}’|^{2}+|\sigma’-\sigma k’|^{2})^{-}
,
\tilde{\sigma}_{k}=[\sigma_{k}’+\beta_{1}(\rho_{k}/2\epsilon_{0})(\sigma’-\sigma_{k}’)]\delta^{-1}
\overline{\eta}_{k}=[\tau_{k}’+\beta_{1}(\rho_{k}/2\epsilon_{0})(\eta’-\tau_{k}’)]\delta^{-1}
\tilde{\mathcal{T}}_{k}=2\epsilon_{0}\beta_{2}(\gamma’/2\epsilon_{0})
,
,
,
\tilde{\tau}_{k}=\tilde{\eta}_{k}-i\tilde{\mathcal{T}}_{k}
where
Then
(
2. 6)’
is chosen to be
\delta
A_{k}^{j}
(\tilde{\tau}_{k}(\tau’, \sigma’),\tilde{\sigma}_{k}(\tau’, \sigma’))\in\overline{\Sigma_{-r}}
Define
A_{k}^{j}
also by (2. 5).
fulfill (2. 3). Remark that
mapps
(i)
(\tilde{\tau}_{k},\tilde{\sigma}_{k})
(ii)
\tilde{\gamma}_{k}\leq 4\epsilon_{0}
(iii)
\tilde{\mathcal{T}}_{k}=T’
(iv)
(\tilde{\tau}_{k},\tilde{\sigma}_{k})=
-into
\overline{\Sigma}
S_{8*_{0}}(\tau_{k}’, \sigma_{k}’)
,
,
for
and
for every
\gamma’\leq 2\epsilon_{0}
(\tau’, \sigma’)
\tilde{\gamma}_{k}\geq 2\epsilon_{0}
for
\gamma’\geq 2\epsilon_{0}
(\tau’, \sigma’)\in S_{2\cdot 0}(\tau_{k}’, \sigma_{k}’)
,
.
according to k are
The reasons of different constructions of
elucidated in \S 5 and \S 7\Gamma
By Corollary 2. 3 of [8] (or c.f . (8. 2)) we obtain the following
such that for
LEMMA 2. 3. Suppose that there exist constants, C,
every
the inequalities
and
(\tilde{\tau}_{k},\tilde{\sigma}_{k})
\mathcal{T}_{0}>0
\gamma\geq\gamma_{0}
(2. 7)
u\in H_{1,\gamma}(R_{+}^{n+1})
||P_{k}^{j}\psi_{k}(D’)u||_{0,\gamma}+|B^{j}\psi_{k}(D’)u|_{z’ r}1\geq C\gamma||\psi_{k}(D’)u||_{0,\gamma}
T. Ohkuho and T. Shirota
92
. Then the estimates (2. 2) are valid.
From Lemma 2. 2 and 2. 3 it is seen that in order to prove (1. 1) for
k=O we have only to obtain the estimate (2. 7) for the localized operators
,
and . Hereafter let
-be one of the points
)
and
be its sufficiently small neighborhood. If a (x,
is a smooth
function of homogeneous degree k in ,
, then we denote by a (x, D’ ) the operator constructed as above, whose
.
symbol
.
is
hold for k=1,
\cdots
,
k_{1}
B^{j}
P_{k}^{j}
(x^{0}, \tau^{0}, \sigma^{0})\in I
\cross\overline{\Sigma}
\{(x^{j},
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\sigma_{k}’)\}
\tau_{\acute{k}}
\tau,
(C^{\infty}(U(x^{0})\cross U(\tau^{0}, \sigma^{0})))
\sigma
(\tau
\sigma)
\tilde{a}(x,
\sigma 1
\tau,
a(\tilde{x},\tilde{\tau},\tilde{\sigma})\cdot
\Lambda_{\gamma}^{k}=a(\tilde{x}(x),\tilde{\tau}(\tau\Lambda_{f}^{1}, \sigma\Lambda_{\gamma}^{1}),\tilde{\sigma}(\tau\Lambda_{\overline{r}}^{1}, \sigma\Lambda_{f}^{-1}))
\Lambda_{r}^{k}\in S_{+}^{k}
\S 3. Decompositions of the problem (P, B) .
into more simple ones, we
In order to decompose the operators ,
by a non-singular and
consider the transformations of the operators ,
m\cross
2m
the eigenvalues of A(x,
matrix. We denote by
smooth 2
(i=1, \cdots, m) the ones with positive and negative
,
and by
, respectively.
imaginary parts for
. Because of the condition (I), in
Let us fix a point
,
we can rearrange the eigenvalues
a neighborhood
into the following three sets I=I_{+}\cup I-, II and III = III III-:
(i= 1, \cdots, l-1) are real and simple at
,
:
,
,
are real and double at
II :
are not real at
to the corresponding sets of
Hereafter we will also identify , II and
indeces \{1,2, \cdots, m\} . It should be noted that some of the sets I , II and
-being considered.
III may be empty according to the point
If the set II is empty, we denote by l the number of elements of the set
P_{k}^{j}
B^{j}
P_{k}^{j}
\lambda
\tau
B^{j}
(x, \tau, \sigma)
\lambda_{i}^{\pm}(x, \tau, \sigma)
\sigma)
(\tau, \sigma)\in C_{-}\cross R^{n-1}
(x^{0}, \tau^{0}, \sigma^{0})\in I
\cross\overline{\Sigma_{-}}
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\{\lambda_{\dot{\sqrt}}^{\pm}(x,
\tau
+\cup
\sigma)\}
I_{\pm}
(x^{0}, \tau^{0}, \sigma^{0})
\lambda_{i}^{\pm}
\lambda_{l}^{+}
III_{\pm}:
(x^{0}, \tau^{0}, \sigma^{0})
\lambda_{l}^{-}
\lambda_{i}^{\pm}(i\geq l+1)
(x^{0}, \tau^{0}, \sigma^{0})
I_{\pm}
III_{\pm}
\subset
\cross\overline{\Sigma}
(x^{0}, \tau^{0}, \sigma^{0})\in I
I_{+}(I_{-})
.
As is shown below, we can take a smooth and non-singular matrix
) defined in
which is homogeneous of degree O in
S (x,
and satisfies
\sigma,
\tau
U(x^{0})\cross U(\sigma^{0}, \tau^{0})
(\sigma, \tau)
S^{-1}P^{0}(x, \tau, \sigma, \lambda)
S=E\lambda-M(x, \tau, \sigma)
where
M=\{\begin{array}{llllll}\lambda_{I}^{+} 0 \lambda_{I}^{-} M_{II} 0 M_{III}^{+} M_{III}^{-}\end{array}\}
,
,
On structures of certain
L^{2}
-well-posed mixed pr oblems for hyperbolic systems
93
\lambda_{1}^{\pm}=\{\begin{array}{lll}\lambda_{1}^{\pm} 0 \ddots 0 \lambda_{l-1}^{\pm}\end{array}\}
,
is a
matrix defined below in Lemma 3. 2 and
are (m-l)\cross
respectively.
m-l) matrices whose eigenvalues are
Here we define the above matrix S. Let
be
) which are homogeneous of degree O in
the eigenvectors of A (x,
, respectively. Then the
and correspond to the eigenvalues { , ,
.
smooth 2 m\cross(l-1) matrices
are analytic in
III_{+}
, ,
, first take all the generalized eigenvectors
As for the set
corresponding to an eigenvalue
with j\in III_{+} . Put
2\cross 2
M_{II}
M_{III}^{\pm}
\{\lambda_{t}^{\pm}\}_{\nu}j\in 1II\pm
’
\{h_{1}^{\pm}, \cdots, h_{l-1}^{\pm}\}=h_{I}^{\pm}
\tau,
(x, \tau, \sigma)
(\tau, \sigma)
\sigma
\lambda_{1}^{\pm}
\cdots
\lambda_{l-1}^{\pm}\rangle
(\tau, \sigma)
h_{I}^{\pm}
h_{1}^{+}
\lambda_{j}^{+}
h_{p_{j}}^{+}
\cdots
(x^{0}, \tau^{0}, \sigma^{0})
h_{fk}^{+}(x, \tau, \sigma)=\frac{1}{2\pi i}\oint cj+(E\lambda-A(x, \tau, \sigma))^{-1}d\lambda\cdot h_{k}^{+}
,
k=1,
\cdots
,
p_{f}
. Rearrange the above
is a small circle enclosing only
(x,
linearly independent vectors
and let them be
, . By the same method as above we can choose vectors
where
c_{j}^{+}
\lambda j
(x^{0}, \tau^{0}, \sigma^{0})
\{h_{l+1}^{+}, \cdots, h_{m}^{+}\}=h_{III}^{+}
\{h_{fk}^{+}\}_{j,k}
\tau
\{h_{l+1}^{-}, \cdots, h_{m}^{-}\}
\sigma)
for III-. Then
homogeneous of degree
For the set II , we
with 2 m components)
=h_{III}^{-}
are
the smooth 2 m\cross(m-l) matrices
0 and analytic in
.
take the smooth 2 m vectors ( i.e . column vectors
h_{III}^{\pm}(x, \tau, \sigma)
(\tau, \sigma)
and
are
such that
,wcseneeow
in
hih i dfid bl
h omogeneousoegreeananaycn
f d
d
lti i
0
Lemma 3. 2.
has the required
. Then
Put
properties, by virture of the linear independence of the column vectors.
Now we describe on the set II in detail.
and be a double root in of
Lemma 3. 1. Let
and
. Then there exist a neighborhood
such that
continuous in
fu nctions
h_{l}’
,
h_{II}’(x, \tau, \sigma)=h_{l}’
h_{II}’(x, \tau, \sigma)=h_{l}’
h_{l}’
()
\tau, \sigma
S(x, \tau, \sigma)
S(x, \tau, \sigma)=(h_{I}^{+}, h_{I}^{-}, h_{II}’, h_{II}’, h_{III}^{+}, h_{III}^{-})
(x^{0}, \tau^{0}, \sigma^{\mathfrak{n}})\in I
\cross
(\overline{\Sigma_{-}}-\Sigma_{-})
\lambda^{0}
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
detP^{0}(x^{0}, \tau^{0}, \sigma^{0}, \lambda)=O
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap(C_{-}\cross R^{n-1}))
\lambda_{II}^{\pm}(x, \tau, \sigma)
(i)
\lambda
{\rm Im}\lambda_{11}^{\pm}(x, \tau, \sigma)>0<
if
{\rm Im}\tau<0
respectively, and
(ii)
\det P^{0}(x, \tau, \sigma, \lambda_{II}^{\pm}(x, \tau, \sigma))=O
if
{\rm Im}\tau\leqq 0
.
Furthermore, they are represented by
(a)
(b)
\lambda_{11}^{\pm}(x, \tau, \sigma)=\lambda_{1}(x, \mu, \sigma)\pm\sqrt{\mu}\lambda_{2}(x, \mu, \sigma)
or
\lambda_{11}^{\pm}(x, \tau, \sigma)=\lambda_{1}(x, \mu, \sigma)\mp i\sqrt{\mu}\lambda_{2}’(x, \mu, \sigma)
according as the normal surface cut by x=x^{0} and
, respectively. Here
cave with respect to at
\tau
(\tau^{0}, \lambda^{0})
\sigma=\sigma^{0}
is convex or con-
T. Ohkubo and T. Shirota
94
\mu=\mu(x, \tau, \sigma) =\tau-\theta(x, \sigma)
is
neighborhood
\theta(x, \sigma)
in
S^{1}
of
(x, \sigma)
(x^{0}, \sigma^{0})
, analytic and real
,
for
\sigma
contained in a conical
,
\theta(x^{0}, \sigma^{0})
=\tau^{0}
,
and
and
in
in a conical neighborhood of
\lambda_{1}\in S^{1}
,
\lambda_{2}’\in S21_{-}
(x, \mu, \sigma)
\lambda_{2}
, analytic and real
(x^{0}, O, \sigma^{0})
whenever
\lambda_{1}(x^{0}, O, \sigma^{0})=\lambda_{0}
,
\lambda_{2}(x^{0}, O, \sigma^{0})>O
,
\mu
for
(\mu, \sigma)
containd
is real,
\lambda_{2}’(x^{0}, O, \sigma^{0})>0
and
\sqrt{\mu}
denotes a branch with positive imaginary part when
\sqrt{1}=-1)
{\rm Im}\mu<0
(i.e. ,
.
COROLLARY 3. 1. In the small neighborhood of
is equivalent to \mu=O , that is,
.
PROOF OF Lemma 3. 1. The strict hyperbolicity implies that
(x^{0}, \tau^{0}, \sigma^{0})
\tau=\theta(x, \sigma)
\lambda_{II}^{-}(x, \tau, \sigma)
\det P^{0}(x, \tau, \sigma, \lambda)=A_{0}(x)\prod_{f=1}^{2m}(\tau-\tau_{j}(x, \sigma, \lambda)
where
is
for every
assume
C^{\infty}
\tau_{j}
\tau_{k}
\lambda_{I1}^{+}(x, \tau, \sigma)=
in
(x, \sigma, \lambda)\in R^{2n+1}
and
j\neq k
, analytic and real for
(x, \sigma, \lambda)\in R^{2n+1}
)
and
. Without loss of generality we may
(\sigma, \lambda)\in R^{n}
\tau_{j}\neq
\tau_{0}=\tau_{1}(x^{0}, \sigma^{0}, \lambda^{0})
1
Since
\lambda_{0}
is a double root it holds that
and
\frac{\partial\tau_{1}}{\partial\lambda}(x^{0}, \sigma^{0}, \lambda^{0})=O
\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x^{0}, \sigma^{0}, \lambda^{0})\neq O
.
Therefore from the implicit function theorem there exists a function
er) which is
C^{\infty}
in
(x, \sigma)
\frac{\partial\tau_{1}}{\partial\lambda}
( x,
and real analytic in
\sigma
,
and
\lambda(x, \sigma))=O
\sigma
such that
\lambda(x^{0}, \sigma^{0})
=\lambda^{0}
.
Hence by Taylor’s expansion we have
\tau_{1}(x, \sigma, \lambda)=\tau_{1}(x,
+ \frac{1}{2}
Since the equation in
\sigma
.
,
\lambda(x, \sigma)
\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}
( x,
(y-\lambda(x, \sigma))
:
)
\sigma
,
\lambda(x, \sigma)
)
(\lambda-\lambda(x, \sigma)
)
2+\cdot
..
,
\lambda(x
On structures
of certain
\tau-\tau_{1}(x,
\sigma
,
L^{2}
-well-posed mixed problems for hyperbolic systems
\lambda(x, \sigma)
= \frac{1}{2}
.
95
)
\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}
( x,
\sigma
,
\lambda(x, \sigma)
)
)
(y-\lambda(x, \sigma) 2+\cdot
..
has real coefficients, it has solutions of the following type:
y-\lambda(x, \sigma)=\zeta^{\frac{1}{2}}+a_{1}(x, \sigma)\zeta^{2}\Sigma+a_{2}(x, \sigma)
where
a_{i}
(x, ) is
\sigma
C^{\infty}
in
\zeta=(\tau-\tau_{1}
Put
(x, \sigma)
( x,
\sigma
,
and real analytic in
\lambda(x, \sigma)
and let
Then note that
\sqrt{\zeta}
(a)
if
2( \frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x,
\sigma
\lambda(x, \sigma)
))-1
\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x^{0}, \sigma^{0}, \lambda^{0})>O
,
\lambda(x, \sigma))^{-1}
}-,
\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x^{0}, \sigma^{0}, \lambda^{0})<0
}
\Sigma 1
\pm\sqrt{\zeta}=\mp i\sqrt{\mu}\{-2(\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x,
\sigma
For
{\rm Im}\tau<0
,
\lambda(x, \sigma))^{-1}
put
y^{\pm}=\lambda(x, \sigma)+(\pm\sqrt{\zeta})+a_{1}(x, \sigma) (\pm.\sqrt{\zeta})^{2}+\cdots
Then
{\rm Im} y^{\pm}={\rm Im}(\pm\sqrt{\zeta})+a_{1}(x, \sigma)
for small
|\zeta|
, respectively.
{\rm Im}(\pm\sqrt{\zeta})^{2}+\cdots<>0
Hence we see
\lambda^{\pm}(x, \tau, \sigma)\equiv y^{\pm}=(\lambda(x, \sigma)+a_{1}(x, \sigma)\zeta+\cdot
\pm\sqrt{\zeta}(1+a_{2}(x, \sigma)\zeta+\cdot
fulfill
(_{\backslash }i)
..)
..)
and (ii). Setting
\theta(x, \sigma)
=\tau_{1}(x,
\sigma
we obtain the desired representation of
,
\lambda(x, \sigma)
\lambda_{II}^{\pm}
),
with
\lambda_{2}(x^{0}, O, \sigma^{0})=\{2(\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x^{0}, \sigma^{0}, \lambda^{0}))^{-1}\}^{1}2>0
or
\zeta
when
{\rm Im}\zeta\leqq 0
\sigma
if
,
\zeta^{1}\Sigma
\pm\sqrt{\zeta}=\pm\sqrt{\mu}\{2(\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x,
(b)
,
\sigma
..
and let
be the positive square root of
be the negative one and its extension when
\mu=\tau-\tau_{1}(x, \sigma, \lambda(x, \sigma))
\zeta>0
))
\zeta^{3}\tau+\cdot
.
T. Ohkubo and T. Shirota
96
\lambda_{2}’(x^{0}, O, \sigma^{\Gamma}’)=\{-2(\frac{\partial^{2}\tau_{1}}{\partial\lambda^{2}}(x^{0}, \sigma^{0}, \lambda^{0}))^{-1}\}^{\frac{1}{2}}>0
according to the case (a) or (b) respectively. Thus the proof is completed.
Hereafter, we will consider mainly only the case (a) of Lemma 3. 1,
because in the case (b) we can treat by the analogous way. Furthermore
, then we denote c (x,
be a function defined in
let c
) with some definition domain U(x^{0})\cross U (O, ).
,
also by c (x,
. Then there exist a neighLemma 3. 2. Let
),
(x,
and a 2\cross 2
and 2m -vectors
borhood
such
and analytic in
) which are smooth in
matrix M_{II} (x,
the following hold:
that for every
are the eigenvector and the
(i)
and
corresponding to
generalized eigenvector of
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
(x, \tau, \sigma)
\mu+\theta(x, \sigma)
\sigma^{0}
\sigma)
\mu,
\sigma
(x^{0}, \tau^{\phi}, \sigma^{0})\in I
\cross
(\overline{\Sigma_{-}}-\Sigma-)
h_{II}’
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\tau,
\tau,
h_{II}’(x, \tau, \sigma)
\sigma
(\tau, \sigma)
(x, \tau, \sigma)
\sigma
(x, \tau, \sigma)\in U(x^{0})\cross U(\tau^{0}, \sigma^{0})
h_{II}’(x, \theta(x, \sigma), \sigma)
h_{II}’(x, \theta(x, \sigma), \sigma)
M(x, \theta(x, \sigma), \sigma)
\lambda_{II}^{-}(x, \theta(x, \sigma), \sigma)
(ii)
(ii)
, respectively.
A(h_{II}’, h_{II}’)=(h_{II}’, h_{II}’)M_{II}
(3. 1)
.
M_{II}(x, \mu, \sigma)
=
where
\lambda_{II}^{+}(x, \theta(x, \sigma), \sigma)=
p_{ij}
(’O,
are smooth in
\sigma
)
\lambda_{1}\Lambda(x,O, \sigma 0(O\sigma)))+\mu
(x, \mu, \sigma)
(\begin{array}{ll}p_{11} p_{12}p_{21} p_{22}\end{array})
, analytic and real
(x, \mu, \sigma)
for real
(\mu, \sigma)
and for
\mu=O
(3. 2)
\Lambda_{\gamma}p_{21}
in the case (a), or
in the case (b).
=\lambda_{2}(x, \mu, \sigma)2>0
\Lambda_{\gamma}p_{21}=-\lambda_{2}’(x, \mu, \sigma)2<0
Furthermore, let
\lambda_{1}(x, \mu, \sigma)=\lambda_{1}(x, O, \sigma)+\lambda_{1}^{(1)}(x, \mu, \sigma)\mu
(3. 3)
\lambda_{1}^{(1)}(x, \mu, \sigma)=2^{-1}(p_{11}+p_{22})
, then
.
is -strictly hyperbolic, there exist
PROOF. Since \det
and a real generalized eigenvector h_{1}(x,
a real eigenvector
,
corresponding to the real double eigenvalue
,
in
and analytic
,
. Note that , are
of
. Put
in , because so is
(E\lambda-A(x, \tau, \sigma))
x_{0}
h_{J}(x, \theta(x, \sigma), \sigma)
\theta
(x, \sigma)
\theta(x, \sigma)
\lambda_{1}(x, O, \sigma)=\lambda_{II}^{\pm}(x
\sigma)
\sigma)
A(x, \theta(x, \sigma), \sigma)
h_{0}
h_{1}
S^{0}
(x, \sigma)
\lambda_{1}(x, O, \sigma)
\sigma
h’(x, \tau, \sigma)
= \frac{1}{2\pi i}\oint_{c_{II}}
where
c_{II}
(
E\lambda-A(x, \tau, \sigma)
)-1h1 (x,
\theta(x, \sigma)
,
is a small circle enclosing only the eigenvalues
\sigma
)
d\lambda
\lambda_{II}^{\pm}(x, \tau, \sigma)
. Then
On structures
of cer tain
L^{2}
97
-well-posed mixed problems for hyperbolic systems
it holds that
h_{II}’
( x,
\theta(x, \sigma)
,
\sigma)=h_{1}
(x,
\theta(x, \sigma)
,
\sigma
)
Hence if we set
\Lambda_{r}h_{II}’(x, \tau, \sigma)=(\lambda_{II}^{\pm}
( x,
\theta(x, \sigma)
,
\sigma)-A(x, \tau, \sigma)
)
h_{II}’(x, \tau, \sigma)
,
(i) is fulfilled. Here we remark that
are linearly independent
. Since (x, )
and analytic in
vectors which are smooth in
\{h_{II}’, h_{II}’\}
(x, \tau, \sigma)
and
h_{II}’(x, \tau, \sigma)
h_{II}’
(\tau, \sigma)
are invariant by the projection operator
\tau,
\frac{1}{2\pi i}\oint_{c_{II}}(E\lambda-A
\sigma
(x,
we see that (ii) is valid.
are real
. Hence
is real if
From Lemma 3. 1
, implies
if
. This together with the analyticity in of
.
that they are always real when
is real and
Therefore it follows from (ii) that
is real. Since (i) and (ii) implies
that defining )
\tau
,
\sigma))^{-1}
dX ,
\tau\geq\theta(x, \sigma)
\lambda_{II}^{\pm}(x, \tau, \sigma)
\{h_{II}’, h_{II}’\}
\tau\geq\theta(x, \sigma)
\{h_{II}’, h_{II}’\}
\tau
(x, \tau, \sigma)\in U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\tau
M_{II}
\Lambda_{0}^{(0}
\equiv\Lambda_{0}(O, \sigma)=
(\theta(x, \sigma)^{2}+|\sigma|^{2})^{\frac{1}{2}}
M_{II}=
(’
\sigma
OO,
)
\lambda_{1}(x,O, \sigma)\Lambda_{0}^{(0)}
)
on \mu=\tau-\theta(x, \sigma)=O , we see that (3. 1) is valid. Hence noting that
eigenvalues of
we have
\lambda_{II}^{\pm}
are
M_{II}
\det
(\begin{array}{ll}\lambda-\lambda_{1}(x,O,\sigma)-p_{11}\mu, -\Lambda_{0}^{(0)}-p_{12}\mu-p_{21}\mu, \lambda-\lambda_{1}(x,O,\sigma)-p_{22}\mu\end{array})
=(\lambda-\lambda_{II}^{+}(x, \mu, \sigma)
)
(\lambda-\lambda_{II}^{-}(x, \mu, \sigma)
)
in the above equation. Then (3. 2) follows from comparing
Put
in the
the coefficient of . Write
left hand side. Then (3. 3) follows from comparing the coefficient of
\lambda=\lambda_{1}(x, O, \sigma)
\lambda-\lambda_{1}(x, O, \sigma)=\lambda-\lambda_{1}(x, \mu, \sigma)+\mu\lambda_{1}^{(1)}(x, \mu, \sigma)
\mu
(\lambda-\lambda_{1}(x, \mu, \sigma))
.
be the extention
Now let
, which is ob) such that
is non-singular and belongs to
of S (x,
tained by the method of \S 2. With fixed j and k put
\tilde{S}(x, \tau, \sigma)=S(\tilde{x}(x),\tilde{\tau}(\tau\Lambda^{-1}\vee’ \sigma\Lambda_{\gamma}^{-1}),\tilde{\sigma}(\tau\Lambda_{\gamma}^{-1}, \sigma\Lambda_{\gamma}^{-1}))
\tilde{S}
\tau,
S_{+}^{J}
\sigma
P_{1}(x, \tau, \sigma, \lambda)=S^{-1}(x, \tau, \sigma)
P_{k}^{j}(x, \tau, \sigma, \lambda)
B_{1}(x’, \tau, \sigma)=B_{j}(x’)\tilde{S}(x’, O, \tau, \sigma)
\tilde{S}(x, \tau, \sigma)
,
,
F(x)=S^{-1}(x, D’)f(x),\cdot
U(x)=S^{-1}(x, D’)u(x) ,
where
S^{-1}
(x, D’ ) is the operator with its symbol
S^{\prime-1}(x, \tau, \sigma)\in S_{+}^{0}
. Then
98
T. Ohkubo and T. Shirota
the problem
(P^{0}, B)
is reduced to the problem
’
(P_{1}, B_{1})J^{P_{1}(x,D)U(x)=F(x)}|B_{1}(x’,D)U(x’, O)=g(x’)
Noting that
the following
Lemma 3. 3.
\tilde{S},\tilde{S}^{-1}\in S_{+}^{0}
inR_{+}^{n+1}onR^{n}.
, we have from corollary 2. 3 of [8] and Lemma 2. 1
(\alpha)
C_{1}
,
\gamma_{1}>0
Let
\psi(\tau, \sigma)
such that for every
(3. 4)
|| P_{1}
\psi(D’)
be
\psi_{k}
\gamma\geq\gamma_{1}
in Lemma 2. 3.
and
If there exist
constants
U\in H_{1,\gamma}(R_{+}^{n+1})
U||_{0,f}+|B_{1}\psi(D’)U|_{\frac{1}{2}r},\geq C_{1}\gamma||\psi(D’)U||_{0,\gamma},\cdot
then there exist C, T_{0}>O such that (2. 7) holds.
, M_{II}(x, D’) and
be the operators of order 1 with
Let
,
,
. Then we
their symbols
and
M_{III}^{\pm}(x, D’)
\lambda_{I}^{\pm}(x, D’)
\tilde{\lambda}_{I}^{\pm}(x, \tau, \sigma)
\overline{M}_{II}(x, \tau, \sigma)
\overline{M}_{II1}^{\pm}(x, \tau, \sigma)
respective,ly\grave{\lrcorner}
see from
(P_{1}, B_{1})
P_{1}(x, D)U=[ED_{n_{\backslash }}-[_{M_{III}^{+}(x,D)}^{\lambda_{I}^{-}(x,D’)}\lambda_{I}^{+}(x,D’),00M_{III}^{-}(x,D’)M_{II}(x,D’)
]
U=F
Hence putting
U(x)={}^{t}(^{t}u_{I}^{+t},u_{I}^{-},\cdot u\text{\’{I}}_{I}, u_{II}^{\prime\prime t},uIII+, tu_{III}^{-})
F(x)=t(fI+, {}^{t}f_{I}^{-}, f_{i_{\acute{I}}}, f_{\acute{I}_{I}’},{}^{t}f_{III}^{+}, f_{III}^{-})
we have for
(x)
(x)
,
x\in R_{+}^{n+1}
,
(P_{I}^{\pm})
P_{I}^{\pm}(x, D)u_{I}^{\pm}=(E_{I}D_{n}-\lambda_{I}^{\pm}(x, D’))u_{I}^{\pm}(x)=f_{I}^{\pm}(x)
(P_{II})
P_{II}(x, D)u_{II}=(E_{II}D_{n}-M_{II}(x, D’))
(P_{III}^{\pm})
P_{III}^{\pm}(x, D)u_{III}^{f}=(E_{III}D_{n}-M_{III}^{\pm}(x, D’))u_{III}^{\pm}=f_{III}^{\pm}(x)
furthermore for
(\begin{array}{l}u_{\acute{I}I}u_{I},\acute{I}\end{array})=(\begin{array}{l}f_{II}’(x)f_{II},,(x)\end{array})
,
,
x’\in R^{n}
B_{1}U=B^{j}(x’)S(x’, O, D’)U(x’, O)=g(x’) ,
(B_{1})
.,
are the unit matrices of order l-1,2 and m-l, respectively. For each above problem we show a priori estimates in \S 5 and
\S 7. Using them we show in \S 8 that the assumptions of Lemma 3. 3 are
fulfilled
where
F_{I}
E_{II}
,
E_{III}
On structures
of certain
L^{2}
-well-posed mixed problems
for
99
hyperbolic systems
\S 4. Lopatinskii determinant, coupling coefficients
and reflection coeficients.
4. 1. Let
) be the matrix defined it some
-and S (x,
which is defined in \S 3. Put
(x^{0}, \tau^{0}, \sigma^{0})\in I
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\cross\overline{\Sigma}
\tau,
\sigma
(V_{I}^{+}, V_{I}^{-}, V_{II}’, V_{II}’, V_{III}^{+}, V_{III}^{-})(x’, \tau, \sigma)=B(x’)S(x’, O, \tau, \sigma)
,
are m -vectors and
wher e
are m\cross(l-1) matrices,
(m-l) matrices.
, where
Let
V_{I1}’
V_{II}’
1^{\gamma_{I}\pm}
S_{II}(x, \mu, \sigma)=(_{s_{21}(x,\mu}^{1}
,
(\overline{C}_{-}\cross Jl^{n-1}))
, because so is
eigenvector of
M_{1I}
\alpha(x, \mu, \sigma)
m\cross
\sigma)
. Then
is continuous in
. Furthermore the first column of
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
S_{II}
S_{II}
is an
\mu=O
. Put
\lambda_{II}^{+}
corresponding to
\lambda_{II}^{+}
,
hence we see that
S_{II}^{-1}M_{II}S_{II}=(\begin{array}{ll}\lambda_{II}^{+} \alpha O \lambda_{1I}^{-}\end{array})
with some
are
s_{21}=(\lambda_{II}^{+}(x, \mu, \sigma)-p_{11}(x, \mu, \sigma)\mu-
o_{1})
\lambda_{1}(x, O, \sigma)) (\Lambda_{0}^{(0)}+p_{12}(x, \mu, \sigma)\mu)^{-1}
V_{III}^{\pm}
which is real for
\mu\geq 0
(x, \mu, \sigma)
and equal to
\Lambda_{0}^{(0}
)
for
S_{1}(x, \mu, \sigma)=(\begin{array}{lll}E_{I} 00 S_{II} E_{III}\end{array})
where
are the 2 (l-1)\cross 2(l-1) , 2 (m-l)\cross
respectively. Then we have from Lemma 3. 2
(4. 1)
E_{I}
,
E_{III}
2(m-l)
identity matrices,
(SS_{1})^{-1}P(SS_{1})=E\lambda-\{\begin{array}{lllll}\lambda_{I}^{+} 0 \lambda_{I}^{-} ----------|\mathfrak{l} \underline{|tOI\lambda_{II}^{-I}|\lambda_{II}^{+}\mathfrak{l}I\alpha|||l}||il|l|I| 0 lt/I_{III}^{+} M_{III}^{-}\end{array})
,
is the eigenvector
This means that the (2l-1) th column of
. Hence
and is homogeneous of degree O in
which corresponds to
putting
we have
SS_{1}
h_{II}^{+}
(\tau, \sigma)
\lambda_{II}^{+}
V_{II}^{+}=Bh_{II}^{+}
BSS_{1}=
B^{+}
DEFINITION 4. 1. Let
, then
and
B^{+} (x’, \tau, \sigma)
(V_{I}^{+}, V_{I}^{-}, V_{II}^{+}, V_{II}’, V_{III}^{+}, V_{III}^{-})
and R(x’, \tau, \sigma)= \det
are called Lopatinskii matrix and
B^{+} (x’, \tau, \sigma)=(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})
R(x’, \tau, \sigma)
.
T. Ohkubo and T. Shirota
100
Lopatinskii determinant of the problem (P, B) for
respectively.
in
are
and
REMARK 4. 1.
and continuous in
, analytic in
. Moreover, if the set II is empty they are
and analytic in
DEFINITION 4. 2. Whm R\neq 0 let us defifine a matrix
(x’, \tau, \sigma)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})
\cap
(\overline{C}_{-}\cross 1^{\supset n-1}\iota))
B^{+} (x’, \tau, \sigma)
R(x’, \tau, \sigma)
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
S_{+}^{0}
(\tau, \sigma)\in C_{-}\cross-R^{n-1}
(C_{-}\cross R^{n-1}))
\sigma^{0})\cap(\overline{C}_{-}\cross R^{n}))
(U(\tau^{0}, \sigma^{0})\cap (\overline{C}_{-}\cross R^{n-1}))
U(x^{0})\cross(U(\tau^{0}
S_{+}^{0}
in
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap (\overline{C}_{-}\cross R^{n-1})
,
U(x^{0})\cross
.
(b_{ij})=(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})^{-1}(V_{I}^{-}, V_{II}’, V_{III}^{-})
and call the elements
Let
b_{if}
coupling
coeffiffifficients with
T_{II}(x, \mu, \sigma)=(\begin{array}{ll}1 \alpha^{-1}(\lambda_{II}^{-}-\lambda_{II}^{+})O 1\end{array})
respect to a matrix
=(T_{II}^{+}, T_{II}^{-})
(SS_{1})
.
.
Then it holds that
T_{II}^{-1}S_{1I}^{-1}M_{II}S_{II}T_{II}=T_{II}^{-1}
(\begin{array}{ll}\lambda_{II}^{+} \alpha O \lambda_{II}^{-}\end{array})
T_{II}=(\begin{array}{ll}\lambda_{II}^{+} OO \lambda_{II}^{-}\end{array})
c
Hence putting
S_{2}(x, \mu, \sigma)=(\begin{array}{ll}E_{I} 0 T_{II}0 E_{III}\end{array})
,
\cdot
we have that (SS_{1}S_{2})^{-1}P(SS_{1}S_{2}) is the matrix (4. 1) with \alpha=O .
which
is the eigenvector
This means that the 2l -th column of
. Hence if we
corresponds to
and is homogeneous of degree 0 in
put
it holds that
SS_{1}S_{2}
h_{II}^{-}
(\tau, \sigma)
\lambda_{II}^{-}
V_{II}^{-}=Bh_{II}^{-}
BSS_{1}S_{2}=
Let
(V_{I}^{+}, V_{I}^{-}, V_{II}^{+}, V_{II}^{-}, V_{III}^{+}, V_{III}^{-})1
(\tilde{b_{if}})=(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})^{-1}(V_{I}^{-}, V_{II}^{-}, V_{III}^{-})
respect to a matrix
SS_{1}S_{2}
be the coupling coefficients with
. Then we have the
is called (generalized) refiection coeffiffifficients with
DEFINITION 4. 3.
, by which A is similar to a generalized diagonal
respect to a matrix
matrix such as (4. 1) with \alpha=O ([3]) .
The reflection coefficients depend, in general, on matrices S, and .
may be considered
the column vectors of
But since for fixed
), for another (generalized)
as eigenvectors or generalised those of A (x,
, there exist non-zero numbers
and corresponding
eigenvectors
and a non-singular matrix T such that
\tilde{b_{ij}}
SS_{1}S_{2}
S_{1}
SS_{1}S_{2}
(x, \tau, \sigma)
\tau,
\langle
\{a_{k}^{\pm}\}
h_{k}^{\pm}\}
\{V_{k}^{\pm}\}
\sigma
S_{2}
On structures of certain
L^{2}
-well-posed mixed problems
for
V_{j}^{\pm}=a_{j}^{\pm}\overline{V}_{j}^{\pm}
j\in I\cup
for hyperbolic
systems
101
II and
,
(V_{l+1}^{+}, \cdots, V_{m}^{+})=(\tilde{V}_{l+1}^{+}, \cdots,\tilde{V}_{m}^{+})T
where
are real if
and
are real. Therefore the invariance of the
reflection coefficients in the following sense is derived from the definition.
depends
,
LEMMA4.1. For fifixed i, j\in I\cup II and
only on the vectors
. In
, where SS_{1}S_{2}=
and
particular, if for some point
are restricted
the vectors
to be real, then
is determined excep t real factor for every i , j\in
a_{j}^{\pm}
\tilde{V}_{f}^{\pm}
V_{j}^{\pm}
(x’, \tau, \sigma)
h_{i}^{+}
b_{ij}^{-}(x’, \tau, \sigma)
(h_{I}^{+}, h_{I}^{-}, h_{II}^{+}, h_{II}^{-}, h_{III}^{+}, h_{III}^{-})
h_{j}^{-}
(x’, \tau, \sigma)
\{h_{1}^{\pm}, \cdots, h_{l}^{\pm}\}
\tilde{b_{ij}}(x’, \tau, \sigma)
IUII
REMARK 4. 2. From the same consideration as in Lemma 4. 1 we see
that the zeroes of Lopatinskii determinant are not depend on the choice
of S and .
Hereafter, taking account of Lemma 4. 1 we choose real vectors
and consider only real
.
whenever
are real, i.e. ,
4. 2. In this subsection we give necessary conditions for the condition
(III) in terms of coupling coefficients. Note that these conditions are also
-well-posed.
sufficient for the constant coefficients problem
to be
THEOREM 4. 1. The condition (III) is equivalent to the following con:
ditions
and
For any
there exist a positive constant C
and a neighborhood
such that for every
(i) for i, j=1, , l
S_{1}
\{h_{I}^{\pm}\}
h_{II}^{\pm}
\mu\geqq 0
\lambda_{II}^{\pm}
(P, B)_{x^{0}}
(\alpha)
L^{2}
(\beta)
(x^{0}, \tau^{0}, \sigma^{c})\in\Gamma\cross
\alpha)
(\overline{\Sigma_{-}}-\Sigma_{-})
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma_{-}
U(\tau^{0}, \sigma^{0})
\cdots
|b_{j}
‘
(ii)
(x^{0}, \tau, \sigma)|\leq C\gamma^{-1}|{\rm Im}\lambda_{i}^{+}(x^{0}, \tau, \sigma)
for i=1,
\cdots
, l and
\cdot{\rm Im}\lambda_{j}^{-}(x^{0}, \tau, \sigma)|^{\frac{1}{2}}
j\in III_{-}
|b_{ij}(x^{0}, \tau, \sigma)|\leq C\gamma^{-1}|{\rm Im}\lambda_{i}^{+}(x^{0}, \tau, \sigma)|^{\frac{1}{2}}
(iii)
for
i\in III_{+}
and j=1,
\cdots
for
i\in III_{+}
for every
,
and j\in III_{-}
|b_{ij}(x^{0}, \tau, \sigma)|\leq C\gamma^{-1}
\beta)
,
,l
|bij(xf, \tau, \sigma)|\leq C\gamma-1|{\rm Im}\lambda_{j}^{-}(x^{0}, \tau, \sigma)|^{\frac{1}{2}}
(iv)
,
r
(x’, \tau, \sigma)\in\Gamma\cross C_{-}\cross R^{n-1}
R(x’, \tau, \sigma)\neq Oo
To show our assertion, first of all we recall a characterization of
L^{2}-
T. Ohkubo and T. Shirota
102
well posedness for constant coefficients case [3], [4]. Let us consider the
constant coefficients problem resulting from freezing the coefficients of P(x,
D) and B(x’) at boundary points x=x^{0} :
n-l
P(D)u(x)=(ED_{n}- \sum_{f=0}A_{f}D_{j}-C)u(x)=f(x)
for x\in Ii_{+}^{n+1}(x_{i)}>0) ,
for x’\in R^{n}(x_{0}>0) ,
for (x’, x_{n})\in R_{+}^{n}(x_{0}=O) ,
(P, B)_{x^{0}}\{
Bu(x’, 0)=0
u (O,
x’,
x_{n}
) =O
and the associated problem by Fourier-Laplace transformation with respect
to
(x’, x_{0})
:
P^{0}(\tau, \sigma, D_{n})
\^u
(\tau, \sigma, x_{n})
= ( ED_{n}-A_{0} \tau-\sum_{f=1}^{n-1} A j a j )
(P^{0}, B)_{x^{0}}\{
=\hat{f}(x_{n})
\theta
for x_{n}>0 ,
B\^u (\tau, \sigma, O)=O .
-well-posed, if and only if
is
Then the problem
that is, there exists a constant C>0 such that for every
has a unique solution
and \hat{f}(x_{n})\in H_{1}(x_{n}>0) ,
which satisfies
(P, B)_{x^{0}}
L^{2}
(P^{0}, B)_{x^{0}}
is so,
(\tau, \sigma)\in C_{-}\cross R^{n-1}
\theta(\tau, \sigma, x_{n})\in H_{1}(x_{n}>0)
(P^{0}, B)_{x^{0}}
(4. 2)
||\theta(\tau, \sigma, \cdot)||\leq C\gamma^{-1}||\hat{f}(\cdot)||
where
||
\^u
( \tau, \sigma, \cdot)||^{2}=\int_{0}^{\infty}|\hat{u}(\tau, \sigma, x_{n})|^{2}dx_{n}
Next to obtain its more concrete characterization, let
-and the solution
arbitrary point. For
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma
B)_{x^{0}}
-be an
,
of
(\tau^{0}, \sigma^{0})\in\overline{\Sigma}
\theta(\tau, \sigma, x_{n})
set
W(\tau, \sigma, x_{n})=(SS_{1})^{-1}(\tau, \sigma)
\^u
(\tau, \sigma, x_{n})
,
where
(SS_{1})(\tau, \sigma)=(h_{I}^{+}, h_{I}^{-}, h_{II}^{+}, h_{II}’, h_{III}^{+}, h_{III}^{-})(\tau, \sigma)
\det SS_{1}(\tau, \sigma)\neq 0
for
,
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}}
If we put
W(\tau, \sigma, x_{n})={}^{t}(^{t}W_{I}^{+},{}^{t}W_{I}^{-}, W_{II}^{+}, W_{II}’,{}^{t}W_{III}^{+},{}^{t}W_{III}^{-})
F(x_{n})=(SS_{1})^{-1}\hat{f}={}^{t}(^{t}f^{++},{}^{t}f_{I}^{-}, fi_{1}+, f_{I}’, f_{III}^{+}, f_{i_{II}^{-}})
(P^{\gamma}(, B)_{x^{0}}
is reduced to problem as follows; for
,
,
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma_{-}
(P^{J}
On structures of certain
L^{2}
-well-posed mixed problems
for
]
(\tau, \sigma)
P_{2}W=[ED_{7l}-\{\begin{array}{lllll}\lambda_{I}^{+} 0 \lambda_{I}^{-} |\lambda II_{I}\alpha_{\mathfrak{l}}^{l}|l--------------- IO\lambda_{II_{l}^{I}}^{I}+^{I}-|||||I|1 0 M_{III}^{+} M_{III}^{-}\end{array}\}
(4. 3)
\{
= F(xn),
x_{n}>0
hyperbolic systems
103
W(\tau, \sigma, x_{n})
.
B_{2}W=(BSS_{1})W
=
(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})
+ (Vt,
V_{II}’
,
\cdot
(\tau, \sigma, O)
{}^{t}(^{t}W_{I}^{+}, W_{II}^{+},{}^{t}W_{III}^{+})
V_{III}^{-}
)
\cdot
(\tau, \sigma, O)=O
t(tW_{I}^{-}, W_{II}’,{}^{t}W_{III}^{-})
.
Setting
W(\tau, \sigma, x_{n})
= (
W_{1,I}^{+},{}^{t}W_{1,I}^{-}
,
W_{1,II}^{+}
,
W_{1,II}’,{}^{t}W_{1}^{+}
,III,
\ell W_{1,III}^{-}
)
+{}^{t}(^{t}W_{2,I}^{+}, O, W_{2,II}^{+}, O,{}^{t}W_{2,III}^{+}, O)
=(W_{1}+W_{2})
(\tau, \sigma, x_{n})
and
W_{1}^{+}
(\tau, \sigma, x_{n})={}^{t}(^{t}W_{1,1}^{+}, W_{1,II}^{+},{}^{t}W_{1,III})
W_{1}^{-}
(\tau, \sigma, x_{n})=
(
W_{1,1}^{-}
,
,
,III),
W_{1,II}’,{}^{t}W_{1}^{-}
W_{2}^{+}(\tau, \sigma, x_{n})={}^{t}(^{t}W_{2,1}^{+}, W_{2,II}^{+},{}^{t}W_{2,III}^{+})
,
W_{2}^{-}(\tau, \sigma, x_{n})=O
,
we consider the following two problems:
\int P_{2}(\tau, \sigma, D_{n})W_{1}(\tau, \sigma, x_{n})=F(x_{n})
(4. 4)
|x_{n}>0
,
W_{1}^{+}(\tau, \sigma, 0)=0
,
,
and
P_{2}(\tau, \sigma, D_{n})W_{2}(\tau, \sigma, x_{n})=O,\cdot
(4. 5)
x_{n}>O
.
\{
W_{2}^{+}(\tau, \sigma, O)+(b_{ij})(\tau, \sigma)W_{1}^{-}(\tau, \sigma, O)=O
Then we see from the uniqueness of the solution of (4. 3) that
satisfying
the solution W of (4. 3) for
From (4. 4) we have
\det(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})(\tau, \sigma)=R\neq O
(\tau, \sigma)
(4. 6)
[ED_{n}-(\begin{array}{lll}\lambda_{I}^{-} 00 \lambda_{II}^{-} M_{III}^{-}\end{array})
W_{1}+W_{2}
(\tau, \sigma)
W_{1}^{-}(\tau, \sigma, x_{n})=(\begin{array}{l}f_{I}^{-}f_{I},\acute{I}\backslash f_{III}^{-}\end{array})
r
is
.
T. Ohkubo and T. Shirota
104
Since
W_{2}^{-}
(\tau, \sigma, x_{n})=O
[ED_{n}-
(4. 7)
\{
(
we have from (4. 5)
\lambda_{I}^{+}O
\lambda_{II}^{+}
)
M_{III}^{+}]0W_{2}^{+}(\tau, \sigma, x_{n})=O
,
x_{n}>0
,
W_{2}^{+}+(b_{ij})W_{1}^{-})(\tau, \sigma, O)=O
From (4. 6) the problem (4. 7) has a unique tempered solution for
{\rm Im}\tau<0
:
W_{2}^{+}(\tau, \sigma, x_{n})
=-\exp
=-\exp
(4. 8)
(iM^{+}(\tau, \sigma)x_{n})\cdot (b_{if})(\tau, \sigma)\cdot W_{1}^{-}(\tau, \sigma, O)
(iM^{+}(\tau, \sigma)x_{n})
\int_{0}^{\infty}\exp
.
(b_{ij})(\tau, \sigma)
(-iM^{-}(\tau, \sigma)x_{n})f’(x_{n})dx_{n}
,
where
M^{\pm}(\tau, \sigma)=(\begin{array}{lll}\lambda_{I}^{\pm} OO \lambda_{11} M_{III}^{\pm}\end{array})
,
f’=(\begin{array}{l}f_{I}^{-}f_{II}’’’\backslash f_{II}^{-}\end{array})
On the other hand, let us put
\tilde{W_{1,II}}=(\begin{array}{l}W_{1,II}^{+}\overline{W_{1},},,II\end{array})\sim
g_{II}=S_{II}^{-1}f_{II}
Then using the relation
becomes for
S_{II}
,
=S_{II}
(\begin{array}{l}W_{1,II}^{+}W_{1}’,\prime II\end{array})=(\begin{array}{l}W_{1,II}^{+}s_{21}W_{1,II}^{+}+W_{1,II}’\end{array})
f_{II}={}^{t}(f_{II}^{+}, f_{II}’)
(\begin{array}{ll}\lambda_{II}^{+} \alpha O \lambda_{II}^{-}\end{array})
S_{II}^{-1}=M_{II}
,
.
, we see that the problem (4. 4)
x_{n}>0 ,
(P_{I}^{+})_{x^{0}}
P_{I}^{+}W_{1.I}^{+}=f_{I}^{+}
,
(P_{I}^{-})_{x^{0}}
P_{I}^{-}W_{1,I}^{-}=f_{I}^{-}
,
(P_{II})_{x^{0}}
P_{II}\cdot t(\overline{W}_{II}^{+},, \tilde{W}_{1,II}’.)
(P_{III}^{+})_{x^{0}}
P_{III}^{+}W_{1}^{+}
(P_{III}^{-})_{x^{0}}
P_{III}^{-}W_{1}^{-}
,III
,III
=f_{III}^{+}
=
f_{III}^{-}
=g_{II}
,
,
,
and
W_{1,I}^{+}(\tau, \sigma, O)=\tilde{W}_{1}^{+}
,II
,III (\tau, \sigma, O)=O
(\tau, \sigma, O)=W_{1}^{+}
.
Hence from Corollary 5. 1 and 7. 1 (ii) described later we have
On structures
of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
||f^{++}||\geq C\gamma||W_{1,1}^{+}||
,
||fI^{-}||\geq C\gamma||W_{1,1}^{-}||
||g_{II}||\geq C\gamma||\overline{W_{1,II}}||
, i.e. ,
||f_{II}||\geq C\gamma(||W_{1,II}^{+}||+||W_{1,II}’||)
||f_{III}^{+}||\geq C\gamma||W_{1,III}^{+}||
,
105
,
||f_{III}^{-}||\geq C\gamma||W_{1,III}^{-}||
,
,
and hence
||W_{1}(\tau, \sigma, \cdot)||\leq C\gamma-1||F(\cdot)||\leq C\gamma-1||\hat{f}(\cdot)||
for
follows that for
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma
Therefore from (4. 2) and
-such that
-.
(\tau, \sigma)\in U(\tau^{0}.
(4. 9)
W_{I}+W_{2}=SS_{I}^{-1}
R(\tau, \sigma)\neq 0
\sigma^{0})\cap\Sigma
||W_{2}(\tau, \sigma, \cdot)||\leq C\gamma^{-1}||\hat{f}(\cdot)||
\mbox{\boldmath$\thea$}
it
,
.
We see from the proof that (4. 9) is also sufficient for
of
.
Put
L^{2}
-well-posedness
(P, B)_{x^{0}}
G_{c}
(x,
y;\tau,
\sigma
)
’/
=e^{iM^{+}(\tau,\sigma)x}(b_{ij})(\tau, \sigma)e^{-iM^{-}(\tau,\sigma)y}
and note that
||f” (\cdot)||^{2}\leq C||\hat{f}(\cdot)||^{2}
=C(||f’(\cdot)||^{2}+||f_{I}^{+}(\cdot)||^{2}+||f_{II}^{+}(\cdot)|^{2}+||f_{III}^{+}(\cdot)||^{2})
.
Then, from (4. 8) and (4. 9) we obtain the following
LEMMA 4. 2. The problem
is -well-posed if and only if the
following )’ and )’ are fulfifilled :
’
For every
there exist a constant C>O and
a neighborhood
such that for every
L^{2}
(P, B)_{x^{0}}
\alpha
\beta
(x^{0}, \tau^{0}, \sigma^{0})\in I
\alpha)
\cross
(\overline{\Sigma_{-}}-\Sigma_{-})
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma_{-}
U(\tau^{0}, \sigma^{0})
\int_{0}^{\infty}|\int_{0}^{\infty}G_{c}(x, y;\tau, \sigma)
f’(y)dy|^{2}dx \leq C^{2}\gamma^{-2}\int_{0}^{\infty}|f’(x)|^{2}dx
,
i.e .,
(4. 10)
\beta)
||G_{c}(x, y;\tau, \sigma)||_{\mathscr{H}(L^{2}} L^{2})\leq C\gamma- 1
,
’
It holds for every
(\tau, \sigma)\in C_{-}\cross R^{n-1}
R(\tau, \sigma)\neq 0
that
.
Thus, in order to prove Theorem 4. 1 we have only to show the
equivalence of the conditions ) and )’. We use the same technique as
in [10].
First note that it holds that
\alpha
(4. 11)
Put
||G,||_{g(L^{2}\cross}
L2,C1)
\alpha
\leq||G_{C}||_{t(L^{2}}\sim
L^{2}
’
)
\leq||G_{c}(x, y)||_{x,y}
.
T. Ohkubo and T. Shirota
106
N_{\pm}=\{\begin{array}{lllll}|Im \lambda_{I}^{\pm}|^{-\frac{1}{2}} 00 |Im \lambda_{II}^{\pm}|^{-_{\tau}^{1}} E_{III}^{\pm}\end{array}\}
.,
L_{+}(x)=N_{+}^{-1}\exp(i^{t}(M^{+})x)
L_{-}(x)=N_{-}^{-1}\exp(-iM^{-}x)
S_{+}= \int_{0}^{\infty}\overline{L_{+}(x)}\cdot
{}^{t}L_{+}(x)
,
,
dx ,
S_{-}= \int_{0}^{\infty}L_{-}(y)\cdot\overline{{}^{t}L_{-}(y)}dy
Then we have
S_{+}N_{+}(b_{ij})N_{-}S_{-}
= \int_{0}^{\infty}\int_{0}^{\infty}\overline{L_{+}(x)}
.
{}^{t}L_{+}(x)
( N_{+}(b_{ij}) N-)
L_{-}(y)\cdot
\overline{{}^{t}L_{-}(y)}
dxdy
= \int_{0}^{\infty}\overline{L_{+}(x)}G_{c}(x, y)\cdot\overline{{}^{t}L_{-}(y)}dxdy
Hence
|S_{+}N_{+} (b_{ij}) N_{-}S_{-}|
(4. 12)
\leq\sup_{f,g\epsilon}|2\int_{0}^{\infty}\int_{0}^{\infty}\langle f(x)
\leq C
,
||G_{c}||_{g(L^{2}\cross}L^{2},C^{1})
G_{c}(x, y)g(y)\rangle dxdy|
,
where . denotes matrix norm and C>0 .
that
|
On the other hand it holds
|
S_{+}=N_{+}^{-1} \int_{0}^{\infty}\exp\{i((M^{+})-((M^{+}))^{*})x\} dxN_{+}^{-1}
0
(2 {\rm Im}\lambda_{I}^{+})^{-1}
=N_{+}^{-1}\{
(2 {\rm Im} \lambda_{II}^{+})^{-1}\int_{0}^{\infty}\exp
(-2{\rm Im} M_{III}^{+}\cdot x)dx\backslash )^{N_{+}^{-1}}
.
0
Since we may assume that
such that for
{\rm Im}(M_{III}^{+})\geq C>0
, there exist constants
C_{1}
,
C_{2}>0
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma
-
C_{1}\leq\overline{S}_{+}=S_{+}\leq C_{2}
(4. 13)
C_{1}\leq S_{-}\leq C_{2}
,
,
where the lower inequalities follow from the same calculation. Combining
(4. 12) and (4. 13) we obtai
On structures of certain
(4. 14)
|N_{+}
L^{2}
-well-posed mixed problems
(bij)
for hyperbolic
N_{-}|\leq C||G_{c}||_{4(L^{2}\cross}L^{2},C^{1})
107
systems
.
On the other hand we have the estimates:
||G_{c}(x, y)||_{x,y}
= \int|\exp (iM^{+}x)(b_{ij})
\exp (-iM^{-}y)|^{2}dxdy
\leq\int|\exp (iM^{+}x)N_{+}^{-1}|^{2}dx\cdot
\leq C|N_{+}
(b_{if})
\int|N_{-}^{-1}
\exp
.
(-iM^{-}y)|^{2}dy\cdot
|N_{+}(b_{if}) N_{-}|^{2}
,
N_{-}|^{2},\cdot
where the last inequality holds for every
with some
C>0 . Combining this with (4. 11) and (4. 14) we see that there exist con, C_{2}>0 such that for every
stants
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma_{-}
C_{1}
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap\Sigma_{-}
C_{1}|N_{+}(b_{ij})N_{-}|\leq||G_{r}||_{\mathscr{H}(}|L^{2},L^{2}),\leq C_{2}|N_{+}(b_{ij})N_{-}|(
Hence (4. 10) is equivalent to
|N_{+} (b_{ij}) N_{-}|\leq C\gamma^{-1}
,
which is equivalent to ). Thus the theorem is proved.
\alpha
\S 5. A Priori estimates for the case where the set II is
empty.
In this section we discuss the decomposed problem
in \S 3 in
the case where the set II is empty for the point
being
considered. In subsection 5. 1 a priori estimates for the problem
and
are given and subsection 5. 2 is devoted to that for the boundary
.
condition
5. 1. For the sake of completeness of our proof of (3. 4) we prove
the following usual
Lemma 5. 1. There exist constants C, r_{0}>0 such that for every
it holds that
(P_{1}, B_{1})
(x^{0}, \tau^{0}, \sigma^{0})\in I
\cross R^{n}
(P_{I}^{\pm})
(P_{III}^{\pm})
(B_{1})
\gamma\geq\gamma_{0}
for every
u_{I}^{\pm}
,
,
(P_{I}^{+})
||PIu^{+}I+||_{0,r}+r^{\neq}|uI+|_{0,r}\geq C\gamma||uI+||_{0}
(P_{I}^{-})
||PI-u_{I}^{-}||_{0,r}\geq C(\sqrt{2}^{1}|u_{I}^{-}|_{0,r}+\gamma||u_{I}^{-}||_{0,r})
,r
(P_{III}^{+})
||P_{III}^{+}u_{III}^{+}||_{0,\gamma}+\gamma|u_{II}^{+}
(P_{III}^{-})
||P_{III}^{-}u_{III}^{-}||_{0,\gamma}\geq C(|u_{III}^{-}|_{\frac{1}{2},\gamma}+||u_{1II}^{-}||1
u_{III}^{\pm}\in H_{1,7}(B_{+}^{n+1})
.
I
I
|_{-_{\tau_{d}}^{1}},\gamma\geq C\gamma||u_{II}^{+}
,
||0
\gamma)
,
,f
,
,
T. Ohkuho and T. Shirota
108
Now we define the norms
||u(
.
||u(
.
)||^{2}= \int_{0}^{\infty}|u(x_{n})|^{2}dx_{n}
and
)||
,
|u|
for
u(x_{u})\in H_{1}(B_{+}^{1},)
by
|u|=|u (0)|
with their inner products denoted by ( , ) and ., , respectively. Then
we have the following estimates for constant coefficients problems, which
are direct consequences of the proof of Lemma 5. 1. and will be used in
the proofs of Theorem 4. 1 and Lemma 6. 3.
and a neighborhood
COROLLARY 5. 1. There exist constants C,
-and
such that for every
\cdot
\langle
\cdot
\cdot\rangle
\gamma_{0}>0
U(\tau^{0}, \sigma^{0})
(\tau\Lambda_{\gamma}^{-1}, \sigma\Lambda_{\gamma}^{-1})\in U(\tau^{0}, \sigma^{0})\cap\Sigma
\gamma\geq\gamma_{0}
(P_{I}^{+})_{x^{0}}
||PI+(x^{0}, \tau, \sigma, D_{n})
\theta_{I}^{+}
(
.
(P_{I}^{-})_{x^{0}}
||P_{I}^{-}(x, \tau, \sigma, D_{n})
\theta_{I}^{-}
(
.
(P_{III}^{+})_{x^{0}}
||P_{III}^{+}(x^{0}, \tau, \sigma, D_{n})\theta_{III}^{+}(\cdot)||+\mathcal{T}\Lambda_{t^{2}}^{-^{1}}|u_{III}^{+}|\geq C\gamma||u_{II}^{+}
(P_{III}^{-})_{x^{0}}
||P_{III}^{-}(x^{0}, \tau, \sigma, D_{n})\theta_{III}^{-}(\cdot)||\geq C(\Lambda_{f}^{1}\tau|u_{III}^{-}|+\Lambda r||u_{III}^{-}(\cdot)||)
for every
\theta_{I}
(x_{n})
,
\theta_{11I}^{\pm}(x_{n})\in H_{1}(R^{1}\mp)
)||+\gamma^{1}r|\theta_{I}^{+}|\geq Cr||\theta_{I}^{+}
. ,
( )||
)||\geq C(\gamma^{\frac{1}{2}}|uI-|+\gamma||\theta_{I}^{-}
. ),
(
)||
I
(\cdot)||
LEMMA 5. 1.
OF
,
.
First we prove the estimates
consider the bilinear form for j\in I and
PROOF
,
(P_{I}^{\pm})
. Let us
u=u_{j}^{\pm}:
2
{\rm Re}
((D_{n}-\lambda_{f}^{\pm}(x, D’))u,
=
{\rm Re}
\mp iu)_{0,\gamma}
\langle\mp u, u\rangle_{0,r}+2
There exists a constant
C_{1}>0
{\rm Re}
(\mp i\lambda_{j}^{\pm}(x, D’)u
such that for any
, u)
r
(x, \tau’, \sigma’)\in U(x^{0})\cross(U(\tau^{0}
\sigma^{0})\cap\overline{\Sigma_{-}})
\mp
Hence from ( 2.
6)’
{\rm Re}
i\lambda_{j}^{\pm}
(x, \tau’, \sigma’)\geq C_{1}\gamma ’
this is also valid for the symbol of
\mp
{\rm Re}
i\lambda_{j}^{\pm}
(\tilde{x},\tilde{\eta}-iT,\tilde{\sigma})
.
\mp i\lambda_{j}^{\neq}.(x, D’)
:
4_{f}\geq C_{1}\tilde{\gamma}\Lambda_{r}
for
, for
=C_{1}(\gamma\Lambda_{f}^{-1})\Lambda_{r}=Clr,
\gamma\Lambda_{\overline{r}}^{1}\leq 2\epsilon_{0}
,
\geq C_{1}2\epsilon_{0}\Lambda_{\gamma}\geq 2\epsilon_{0}C_{1}\gamma
\mathcal{T}\Lambda_{\overline{r}}^{1}\geq 2\epsilon_{0}
,
\{
\geq C\gamma
for every
(x, \tau, \sigma)\in\overline{R_{+}^{n+1}}\cross C_{-}\cross
||
(
D_{n}-\lambda_{j}^{\pm}(x, D’)
)
R^{n-1}
Using Lemma 2. 1
u||_{0,r}\cdot||u||_{0}
for large . Accordingly for any
\gamma
.
,
\delta>O
\beta
)
(v)
\geq C(\mp|u|_{0,\gamma}^{2}+\gamma||u||_{0,r}^{2})
\tau
we have
we have
,
On structures of certain
\frac{1}{\delta\gamma}||
L^{2}
(
-well-posed mixed problems for hyperbolic systems
D_{n}-\lambda_{i}^{\pm}(x, D’)
)
109
u||_{0,r}^{2}+\delta\gamma||u||_{0,r}^{2}
\geq C(\overline{\perp}|u|_{0,r}^{2}+\gamma||u||_{0,r}^{2})
Choosing sufficiently small we obtain the inequalities
.
Next we prove the estimates
. Consider the bilinear form for
:
(P_{III}^{\pm}(x, D)u,
2
\delta
(P_{I}^{\pm})
(P_{III}^{\pm})
u=u_{III}^{\pm}
{\rm Re}
\mp i\Lambda u)_{0,r}
=\mp
where
\Lambda^{s}u=
{\rm Re}
\langle u, \Lambda u\rangle_{0,r}+
(2 \pi)^{-n}\int_{R^{n}}e^{i.x’}\underline{.}’|\xi’|^{s}
\mp iM_{III}^{\pm}(x, D’)
2{\rm Re}
(\mp iM_{III}^{\pm}(x, D’)u
,
\Lambda u
and the symbol
\^u (\xi’, x_{n})d\xi’
)
,f
\mp i\overline{M_{III}^{\pm}}(x, \tau, \sigma)
of
is
\mp iM_{III}^{\pm}(\tilde{x},\tilde{\tau},\hat{\sigma}.)
Since
assume
or
{\rm Im}\tau^{0}\leq-3\epsilon_{0}
{\rm Im}\tau^{0}=O
.
\Lambda_{r}
.
, taking sufficiently small
we may
with C>O ,
\pm i(M_{III}^{\pm}-(M_{III}^{\pm})^{*})(x, \tau’, \sigma’)\geq C
for every \{x,
.
If we are considering such a point
then the Hermite part of
is
\tau’,
U(\tau_{0}, \sigma_{0})
\sigma’)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}})
-that
(x^{0}, \tau^{0}, \sigma^{0})\in\Gamma\cross\overline{\Sigma}
{\rm Im}\tau_{0}\leq-3\epsilon_{0}
,
\mp iM_{III}^{\pm}(\tilde{x},\tilde{\tau},\tilde{\sigma})
(5. 1. 1)
\overline{\perp}i(M_{III}^{\pm}-(M_{III}^{\pm})^{*})(\tilde{x}(x),\tilde{\tau}(\tau\Lambda_{\gamma}^{1}, \sigma\Lambda_{\gamma}^{1})
for every { x,
\sigma)\in\overline{B_{+}^{n+1}\prime}\cross\overline{C_{+}’}\cross
\tau,
R^{n-1}
,
\tilde{\sigma}(\tau\Lambda_{\overline{r}}^{1}, \sigma\Lambda_{\overline{r}}^{1}))\geq C
, because from
2.
we have
, which is one reason for dif(
4)^{\ell}
(\tilde{\tau},\tilde{\sigma})\in
and then
ferent constructions of
i n \S 2.
If a point
with
is being considered, then
(5. 1. 1) is also valid, because from ( 2. 6’) we have
S_{\frac{6}{2}\epsilon_{0}}
(\tau^{0}, \sigma^{0})\subset U(\tau^{0}, \sigma^{0})
{\rm Im}\tilde{\tau}\leq-_{\mathfrak{T}}^{1}\epsilon_{0}
A_{k}^{f}
(x^{0}, \tau^{0}, \sigma^{c})\in I
{\rm Im}\tau^{0}=O
\cross\overline{\Sigma_{+}}
(\tilde{\tau},\tilde{\sigma})\in S_{8\epsilon_{0}}(\tau^{0}, \sigma^{0})\subset U(\tau^{0}, \sigma^{0})
for small .
Since
\epsilon_{0}
\mp iM_{III}^{\pm}(\tilde{x},\tilde{\tau},\tilde{\sigma})\in S_{+}^{0}
we obtain from Lemma 2. 1 ) (i)
\beta
with C>0
\frac{1}{\delta}||P_{III}^{\pm}u||_{0,r}^{2}\geq\mp|u|_{1}^{2},r+C||\Lambda u||_{0,r}^{2}\tilde{2}
if
\delta>0
is chosen sufficiently small. On the other hand
D_{n}u=P_{III}^{\pm}u+M_{III}^{\pm}u
Hence it holds that
||D_{n}u||_{0,r}^{2}\leq C
Thus we obtain
(
||P_{III}^{\pm}u||_{0,r}^{2}+||\Lambda
u
||_{0}^{2}
,r
).
T. Ohkubo and T. Shirota
110
||P_{III}^{+}u||_{0}^{2}
,
\gamma+|u|_{\frac{21}{2}}
, f\geq C
||u||_{1}^{2}
,r
and
||P_{III}^{-}u||_{0,r}^{2}\geq C(||u||_{1,\gamma}^{2}+|u|_{\tau}^{2}1,\gamma)
.
This proves
Let us show the inequality
(P_{III}^{-})
.
.4
2{\rm Re}(P_{III}^{\pm}(x, D)u,
(P_{III}^{+})
. Consider the bilinear form
\overline{\perp}i\Lambda^{-1}u)_{c,r}
=
{\rm Re}
\langle u, \overline{\perp}\Lambda^{-1}u\rangle_{0,f}+2{\rm Re}(\mp iM_{III}^{\pm}(x, D’)u,
=
{\rm Re}
\langle u, --_{I}-\Lambda^{-1}u\rangle_{0,r}+2{\rm Re}(\mp iM_{III}^{\pm}(x, D’)\Lambda_{\mathfrak{l}}^{-1}u
+ 2{\rm Re}
(
u, u
[\mp iM_{III}^{\pm}, \Lambda^{-1}]
4^{-1}u)_{0,f}
,
u)_{0,\gamma}
).
Then by the same way as in the proof of
(P_{III}^{-})
, we see
that
\delta^{-1}\gamma^{-2}||P_{III}^{\pm}u||_{0,r}^{2}+\delta T^{2}||\Lambda^{-1}u||_{0,\gamma}^{2}
\geq\mp|u|_{-\frac{1}{2}}^{2}
,
\gamma+C
||u||_{0}^{2}
with C<0 .
,f
and complete the proof of Lemma
we obtain
Since
,
5. 1.
5. 2. Let us consider a priori estimates for the boundary condition
in the case where the set II is empty. If we use the notations in \S 4
is rewritten by
\mathcal{T}^{2}||\Lambda^{-1}u||_{0}^{2}
(P_{III}^{+})
\gamma\leq||u||_{0,/}^{2}
(B_{1})
(B_{1})
(B_{1})
(V_{I}^{+}, V_{I}^{-}, V_{III}^{+}, V_{III}^{-})(x’, D’)\cdot t(tu_{I}, {}^{t}u_{I}^{-t},u_{III}^{+}, tu_{III}^{-})
=g (x’)
in
R^{n}
(x’, O)
,
where the symbol of
. Note that in this case it holds that
(V_{I}^{+}, V_{I}^{-},\cdot V_{III}^{+}, V_{III}^{-})(x’, D’)
is
(V_{I}^{+},\cdot V_{I}^{-}, V_{III}^{+}, V_{III}^{-})(\tilde{x},\tilde{\tau}
,
\tilde{\sigma})
B^{+} (x’, \tau, \sigma)=(V_{I}^{+}, V_{III}^{+})
R(x’, \tau, \sigma)=
hence
(V_{I}^{+}, V_{III}^{+})
and
R(x’, \tau, \sigma)
S_{+}^{0}(U(x^{0})\cross U(\tau^{0}, \sigma^{0}))
(B_{1})
the following
THEOREM 5. 1. Assume the conditions (II)
is empty, there exist constants C,
\gamma_{0}>0
and (III). If the set II
such that for every
and
|B^{+}(x’, D’)
\geq C
(
v^{+}(x’)|_{\tau}1,r
\gamma^{1}z|v_{I}^{+}|_{0,\gamma}+\gamma
|vIII+|_{-\frac{1}{2}}
\alpha)
\gamma\geq\gamma_{0}
v^{+}(x’)={}^{t}(^{t}v_{I}^{+},{}^{t}v_{III}^{+})\in H\tau l,r(R^{n})
(B^{+})
,
and analytic in
are in
(C-\cross R^{n-1}) .
Then an a priori estimate for
is given by
B^{+} (x’, \tau, \sigma)
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap
\det
,r
)
r
On structures of certain
L^{2}
-well-posed mixed problems for hypei’bolic systems
111
PROOF OF THEOREM 5. 1. If
the theorem is a direct
consequence of Lemma 2. 1 ) (ii) . Hence we may assume from the condition (III) and Theorem 4. 1 that
.
for
Since
is analytic in , the condition (II)
implies
R(x^{0}, \tau^{0}, \sigma^{0})\neq 0
\beta
R(x^{0}, \tau^{0}, \sigma^{0})=O
R(x^{0}, \tau, \sigma)
\alpha)
\tau
(5. 1)
(x^{0}, \tau^{0}, \sigma^{0})\in\Gamma\cross’ R^{n}
C\gamma\leq|R(x^{0}, \tau^{0}-i\gamma, \sigma^{0})|\leq C’\gamma_{:}
with some C, C’>0 . Hence we have
(5. 2)
\frac{\partial R}{\partial\tau}(x^{0}, \tau^{0}, \sigma^{0})\neq 0
Since
degree 0 in
R(x’, \tau, \sigma)
(\tau, \sigma)
(5. 3)
in some
is
in
.
and homogenous of
, we have from the implicit function theorem
C^{\infty}
(x’, \tau, \sigma)
, analytic in
R(x’, \tau, \sigma)=(\tau-\nu(x’, \sigma)
).
(\tau, \sigma)
c_{0}(x’, \tau, \sigma)
where
and
are smooth,
O,
and
are homogeneous of degree 1 in and
is
homogeneous of degree -1 in
. From Theorem 4. 1 it holds that
R(x’, \tau, \sigma)\neq O for {\rm Im}\tau<O .
Hence we have
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\nu_{1}(x, \sigma)
,
\nu=\nu_{1}+i\nu_{2}
c_{0}(x^{0}, \tau^{0}, \sigma^{0})\neq
c_{0}
\nu_{2}(x, \sigma)
\sigma
c_{0}(x’, \tau, \sigma)
(\tau, \sigma)
(5. 4)
\nu_{2}(x’, \sigma)\geq O
.
To complete the proof of the theorem we require the following two
lemmas.
LEMMA 5. 2. Under the same conditions as in Theorem 5. 1, there
exists a neighborhood
such that the following hold:
(i) There exist indeces j\in III_{+} and k\in III-such that for any
the vectors
are linearly independent.
Let j\in III_{+} in (i) be l+ l, for simplicity, and denote by
the linear
space spaned by the vectors in the parenthesis. Then we have for
such that
(ii)
, and
(iii)
for every
PROOF OF LEMMA 5. 2.
(i) First of all we remark that
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
(x, \tau, \sigma)
\in U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\{V_{I}^{+}.
V_{l+1}^{+}, \cdots, V_{j-1}^{+}, V_{k}^{-}, V_{f+1}^{+}, \cdots, V_{m}^{+}\}
L(\cdot)
(\tau_{@}, \sigma)\in
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\tau=\nu(x^{0}, \sigma)
V_{l+1}^{+}(x^{0}, \tau, \sigma)\in L(V_{l+2}^{+}, \cdots, V_{m}^{+})
V_{j}^{-}(x^{0}, \tau, \sigma)\in L(V_{I}^{+}, V_{l+2}^{+}, \cdots, V_{m}^{+})
(5. 2. 1)
for every j,
\sum_{f=1}^{m}\det(V_{1}^{+}
k\in I_{+}\cup III_{+}
,
\cdots
,
\frac{\partial V_{f}^{+}}{\partial\tau}
,
\cdots
,
\cdot
j\in I_{-}
V_{m}^{+})(x^{0}, \tau^{0}, \sigma^{0})\neq 0
.,
T. Ohkubo and T. Shirota
112
(5. 2. 2)
\det(V_{1}^{+}, \cdots,\check{V}^{++}, \cdots, V_{m}^{+})(x^{0}, \tau f, \sigma)=O
(5. 2. 3)
rank
,
(V_{I}^{+}, V_{III}^{+}, V_{I}^{-}, V_{III}^{-})(x’, \tau, \sigma)=m
in some
which follow from (5. 2), (5. 3) and (I)
Suppose that there exists an index
(5. 2. 4)
\tau=\nu(x^{0}, \sigma)
on
\det(V_{1}^{+},
\cdots
,
\frac{\partial V_{j}^{+}}{\partial\tau}
,
\cdots
,
\gamma)(ii)
U(x^{\cap})\cross U(\tau^{0}, \sigma^{0})
,
, respectively.
such that
j\in I_{+}
V_{m}^{+})(x^{0}, \tau,\sigma)\neq 0
Then we first show it contradicts (5. 2. 3) under the condition (III).
such that for
From Theorem 4. 1 there exists a neighborhood
U_{1}(\tau^{0}, \sigma^{0})
(\tau’, \sigma’)\in U_{1}(\tau^{0}, \sigma^{0})\cap\Sigma_{-}
(5. 2. 5)
|\det
(V_{1}^{+}, \cdots, \grave{V}_{k}^{-}j\nearrow, \cdots, V_{m}^{+})(x^{0}, \tau’, \sigma’)|
\leq C(x^{0}, \tau^{0}, \sigma^{0})
for
j\in I_{+}
U(\tau^{0}, \sigma^{0})
.
|{\rm Im} \lambda_{j}^{+}(x^{\eta}, \tau’, \sigma’)|^{\frac{1}{2}}\cdot|R(x’, \tau’, \sigma’)|\cdot(\mathcal{T}’)^{-1}
. On the other hand, there exists a neighborhood
and
such that for any
k\in I_{-}\cup III_{-}
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap(C_{-}\cross R^{n-1})
(\tau\Lambda_{f}^{-1}, \sigma\Lambda_{\overline{r}}^{1})\in U_{1}(\tau^{0}, \sigma^{\eta})\cap\Sigma_{-}
.
and R are homogeneous of order 0 and
Since
order 1, we have from (5. 2. 5)
V_{t}^{\pm}
\lambda_{j}^{\pm}
homogeneous of
j
|\det
(V_{1}^{+}, \cdots, \check{V}_{k}^{-}, \cdots, V_{m}^{+})(x^{0}, \tau, \sigma)|
\leq
C(x^{0}, \tau^{0}, \sigma^{0})|{\rm Im}\lambda_{j}^{+}(x^{0}, \tau, \sigma)|^{1}\tau
\cross|R(x^{0}, \tau, \sigma)|.\gamma^{-1}.(|\tau|^{2}+|\sigma|^{2})^{\frac{I}{2}}
for
it is seen that for every
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap
(C_{-}\cross R^{n-1})
. Note that
in
and small T>0
|\tau|^{2}+|\sigma|^{2}\leq C
(\eta, \sigma)\in U(\tau^{0}, \sigma^{0})\cap R^{n}
U(\tau^{0}, \sigma^{0})
. Then
j
(5. 2. 6)
|\det
(V_{1}^{+}, \cdots, \check{V}_{k}^{-}, \cdots, V_{m}^{+})(x^{0}, \eta-i\gamma, \sigma)|
\leq C
.
|{\rm Im} \lambda_{j}^{+}(x^{0}, \eta-i\gamma, \sigma)|^{1}\tau
with some C>0 . Since
(5. 2. 7)
|{\rm Im}
(
\lambda_{f}^{+}
.
|R(x^{0}, r,-i\gamma, \sigma)|\gamma^{-1}
is a simple root we have
\lambda_{j}^{+}(x^{0}, \eta-i\gamma, \sigma)|\leq C\gamma
for
j\in I_{+}
in (5. 2. 6), (5. 2. 7) and make
as
with some C>0 . Take
converge to zero. Then it follows from (5. 1) that
(\tau^{0}, \sigma^{0})
(5. 2. 8)
(\eta, \sigma)
\det(V_{1}^{+}, \cdots, \dot{V_{k}}^{-}j, \cdots, V_{m}^{+})(x^{9}, \tau^{0}, \sigma^{0})=0
\gamma>O
On structures of certain
L^{2}
113
-well-posed mixed problems for hyperbolic systems
.
and
for every
On the other hand (5. 2. 4) implies
k\in I_{-}\cup III_{-}
j\in I_{+}
\{V_{1}^{+}, \cdots, V_{j-1}^{+}, V_{j+1}^{+}, \cdots, V_{m}^{+}\}(x^{0}, \tau^{0}, \sigma^{0})
are linearly independent.
such that
Hence from (5. 2. 8) there exist numbers
V_{k}^{-} (x^{0}, \tau^{0}, \sigma^{0})=\sum_{i\neq j}^{m}a_{i}^{k}V_{i}^{+}
for every
k\in I_{-}\cup III_{-}
\{a_{i}^{k},\}
.
This together with (5. 2. 2) contradicts (5. 2. 3). Hence we see from (5. 2. 1)
that there exists an index j\in III_{+} satistying (5. 2. 4).
Secondly, by the same method deriving (5. 2. 8) from (5. 2. 5), we see
.
from Theorem 4. 1 (iii) that (5. 2. 8) is valid for every j\in III_{+} and
This together with (5. 2. 2) and (5. 2. 3) implies (i).
(ii) If the set I is empty, the lemma follows immediately, because
are linearly independent and from (5. 3)
from (i)
. Hence we
are linearly dependent on
,
,
are lineassume that the set I is not empty and show that {
k\in I_{-}
\{V_{l+2}^{+}, \cdots, V_{m}^{+}\}(x’, \tau, \sigma)
\{V_{l+1}^{+}, \cdots, V_{m}^{+}\}
\tau^{=}1j(x^{0}, \sigma)
(x’, \tau, \sigma)
V_{l+1}^{+}
.
arly dependent on
Here we use again (5. 2. 6).
\cdots
V_{m}^{+}\rangle
\tau=\nu(x^{0}, \sigma)
\nu_{1}(x^{0}, \sigma)
Take
, then it follows from (5. 3) and the condition (II)
(5. 2. 9)
(\eta, \sigma)\in U(\tau^{0}, \sigma^{0})\cap R^{n}
\alpha)
such that
that
\eta=
|R(x^{0}, \eta-i\gamma, \sigma)|
\leq C|\nu_{1}(x^{0}, \sigma)-i\gamma-\nu(x^{0}, \sigma)|=C\gamma
Making
\gamma>0
converge to zero we obtain from (5. 2. 6), (5. 2. 7) and (5. 2. 9)
\det(V_{1}^{+}, \cdots,\check{V}_{k}^{-}i, \cdots, V_{m}^{+})(x^{0}, \eta, \sigma)=O
(5. 2. 10)
for
.
such that
-and
In view of (5. 2. 3) we see from (5. 2. 10) and (5. 2. 2) that for every
j\in I_{+}
,
k\in I_{-}\cup III
\eta=\nu 1(x^{0}, \sigma)=\nu(x^{0}, \sigma)
(\eta, \sigma)\in U(\tau^{0}, \sigma^{0})\cap R^{r_{u}}
\{v_{i}\}\subset C^{m}
\det(v_{1}, \cdots, v_{l}, V_{III}^{+})=O
on
\tau=\nu\{xf
are linearly dependent on
{ xf,
This means that
(iii) As in (5. 2. 10) we see from Theorem 4. 1 (iii)
\tau=\nu
\{V_{III}^{+}\}
(5. 2. 11)
\det(V_{I}^{+}, \mathcal{V}_{i}^{-}, V_{l+2}^{+}, \cdots, V_{m}^{+})(x^{0}, \tau, \sigma)=O
\sigma)
,
\sigma)
\in R^{1}
.
and proves (ii).
i\in I_{-}
,
,
,
such that
{ xf, . Since from (i) { ,
are lineafly independent, the statement (iii) follows from (5. 2. 11)
directly.
LEMMA 5. 3. Under the same conditions as in Theorm 5. 1 the fof-
for
V_{m}^{+}\}
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap R^{n}
\tau=\nu
\sigma)
V_{I}^{+}
V_{l+2}^{+}
\cdots
T. Ohkubo and T. Shirota
114
lowing hold :
(i) There exist smooth scalar fu nctions
tor fu nction a(x’, \tau, \sigma) which are homogeneous
respectively, and satisfy
and a smooth vec,
order O and -1 in
a_{i}(x’, \tau, \sigma)
of
(\tau, \sigma)
V_{l+1}^{+}-a_{l+2}V_{l+2}^{+}-\cdots-a_{m}V_{m}^{+}=(\tau-\nu(x’, \sigma)
in
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
). a
.
There exists a smooth
such that
order 0 in
(ii)
m\cross l
matrix
C_{I}(x’, \tau, \sigma)
homogeneous
of
(\tau, \sigma)
V_{I}^{-}
(x’, \tau, \sigma)=B^{+}\cdot C_{I}
in
U(x^{0})\cross U(\tau^{0}, \sigma^{0})\downarrow
LEMMA 5. 3. Before proving it should be noted that Lemma
5. 2 (ii), (iii) are valid if \tau= lj(x0, ) is replaced \tau= v\{x’ , ), because in their
proofs we can take
as an arbitrary point such that the condition (II)
.
holds for some
(i) From Lemma 5. 2 (i) and (ii) it holds that
PROOF
OF
\sigma
\sigma
x^{0}
\alpha)
(\tau^{0}, \sigma^{0})
(5. 3. 1)
rank
(V_{l+1}^{+}, V_{l+2}^{+}, \cdots,\cdot V_{m}^{+})=m-l-1
rank
(V_{l+2}^{+}
on
\tau=\nu\{xf
,
\sigma)
and
(5. 3. 2)
To determine
a_{f}(x’, \tau, \sigma)
(5. 3. 3)
,
\cdots
.
in
V_{m}^{+})=m-l-1
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
.
we consider the system of equations:
(V_{l+2}^{+}, \cdots, V_{m}^{+})
(\begin{array}{l}a_{l+2}\vdots a_{m}\end{array})=V_{l+1}^{+}(x’, \tau, \sigma)
.
Since there is such a minor of order m l-1 of the matrix
that is non-singular, let it be the upper part of that matrix and
write it by
(V_{l+2}^{+}, \cdots, V_{m}^{+})
–
(x^{0}, \tau^{0}, \sigma^{0})
C(x’, \tau, \sigma)=(\begin{array}{llll}v_{l+}^{+}.2,1 \prime- \ldots, v_{m}^{+}.,1v_{l+}^{\dotplus}.2,m -l-1, \ldots, v_{m}^{\dotplus}..m-l-1\end{array})
where
\det C\neq 0
in some
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
(\begin{array}{l}a_{l+2}^{\backslash }\vdots a_{m}\end{array})=C^{-1}
.
(\begin{array}{l}v_{l+1,1}^{+}\vdots\backslash ^{v_{l+1,m-l-1}^{+}}\end{array})
,
Put
(x’, \tau, \sigma)
.
, analytic and homogeneous of degree 0, with
in
, and fulfill the upper m-l-l equations of (5. 3. 3).
respect to
Moreover, we see that they are also the solution of the system when
by virture of (5. 3. 1) and (5. 3. 2)
Then
\{a_{i}\}
are
C^{\infty}
(\tau, \sigma)
\tau=\nu(x’, \sigma)
(x’, \tau, \sigma)
On structures of certain
L^{2}
-well-posed mixed problems
for
115
hyperbolic systems
Put
f(x’, \tau, \sigma)=V_{l+1}^{+}-a_{l+2}V_{l+2}^{+}-\cdots-a_{m}V_{m}^{+}={}^{t}(f_{1}, \cdots, f_{m})
Then
and
f is
C^{\infty}
in
(x’\tau, \sigma)
.
, analytic and homogeneous of degree 0 in
on
f(x’, \tau, \sigma)=O
U(x^{0})\cross U(\tau^{0}, \sigma^{0})\cap\{\tau=\nu(x’, \sigma)
(\tau, \sigma)
}
f
Apply Wierstrass preperation theorem to . Then there exist integers
which are analytic and homogeneous
and smooth functions
and satisfy for j=1, , m
of order -m_{f} i n
g_{i}(x’, \tau, \sigma)
\geq 1
(\tau, \sigma)
m_{j}
\cdots
f_{j}(x’, \tau, \sigma)=(\tau-\nu(x’, \sigma)
) mj .
g_{j}(x’, \tau, \sigma)
,
in some
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
.
If we put
a(x’, \tau, \sigma)=(t(\tau-\nu)^{m_{1}-1}g_{1},
\cdots
,
(\tau-\nu)^{m_{m}-1}g_{m})
,
the statement (i) is proved.
(ii) By the same way as in the proof of (i) it follows from Lemma
and smooth
5. 2 (iii) that there exist smooth scalar functions
O
and -1 in
which are homogeneous of order
vector functions
respectively, and satisfy
a_{Ij}(x’, \tau, \sigma)
d_{t}(x’, \tau, \sigma)
(\tau, \sigma)
(5. 3. 4)
V_{i}^{-}-(a_{i,1}V_{1}^{+}+\cdots+a_{i} lV_{l}^{+}+a_{i,l+2}V_{l+2}^{+}+\cdots+a_{i} , mmV^{+}
,
=(\tau-(x’, \sigma)
for i=1,
\cdots
, l.
)
in some
d_{i}
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
Put
D_{1}(x’, \tau, \sigma)=\{\begin{array}{lll}a_{1,1} \cdots a_{m,1}\vdots \vdots a_{1,l} a_{m,l}O \cdots Oa_{1,l+2} a_{m,l+2}\vdots \vdots a_{1,n} \cdots a_{m,m}\end{array}\}
,
D_{2}=(d_{1}, \cdots, d_{l})
.
Then we have from (5. 3. 4)
V_{I}^{-}=B^{+}D_{1}+(\tau-\nu(x’, \sigma)
)
D_{2}
,
,
are homogeneous of order O , -1 in
where
it holds that
any non-singular m\cross m matrix
D_{1}
D_{2}
C_{1}
(\tau-\nu)D_{2}
=C_{1}(\tau-\nu)C_{1}^{-1}D_{2}
(\tau, \sigma)
respectively. For
T. Ohkubo and T. Shirota
116
=C_{1}(\begin{array}{lll}E_{1}^{+} 00 (\tau-\nu)\Lambda_{\gamma}^{-1} E’\end{array})
(
=C_{1} \{ E_{I}^{+}00E’(\tau-\nu)\Lambda_{\gamma}^{-1}
)(
)’
E_{I}^{+}00E’(\tau-\nu)\Lambda_{\gamma}^{-1})^{-1}(\tau-\nu) C_{1}^{-1}D_{2}
(\tau-\nu)E_{I}^{+}0^{\backslash }0_{(\tau-\nu)E’}^{\Lambda_{\gamma})C_{1}^{-1}D_{2}}
\backslash
where
, E’
E_{I}^{+}
are the
l\cross l,
(m-l-1)\cross(m-l-1) identity matrices res-
pectively.
\tau
,
\sigma)
We will choose
in (i) and put
C_{1}
in order to show that
C_{0}(x’, \tau, \sigma)=\{\begin{array}{lllllllllll}E_{I}^{+} | 0 | ---- --- ------- -\cdot-- -\cdot\cdot --- .--- \cdots !
V_{I}^{-}\in rangeB^{+}
. Take
a_{i}(x’,\cdot
1 0 I 0 ||| -a_{l+2} \vdots I -a_{m} 0 1\end{array}\}
1
Then with
a(x’, \tau, \sigma)
in (i) it holds that
B^{+}C_{0}(x’, \tau, \sigma)
=(V_{I}^{+}, (\tau-\nu)a,
V_{l+2}^{+}
,
\cdots
,
V_{m}^{+})
(5. 3. 5)
=(V_{I}^{+}, a\Lambda_{\gamma}, V_{l+2}^{+}, \cdots, V_{m}^{+}) (_{0}^{E_{I}^{+}}(\tau-\nu(x’, \sigma)
Since
\det C_{0}(x’, \tau, \sigma)\equiv 1
)
\Lambda_{\gamma}^{1}E0
,)
1
we have
R=\det B^{+}=\det B^{+}C_{0}
= ( \tau- l) )
\det(V_{I}^{+}, a, V_{l+2}^{+}, \cdots, V_{m}^{+})
,
where from (5. 2)
\det
(V_{I}^{+}, a, V_{l+2}^{+}, \cdots, V_{m}^{+})\neq O
.
Hence if we put
C_{1}=(V_{I}^{+}, a\Lambda_{f}, V_{l+2}^{+}, \cdots, V_{m}^{+})
is homogeneous of order 0 in
then
we obtain from (5. 3. 5)
C_{1}(x’, \tau, \sigma)
(\tau, \sigma)
V_{I}^{-}=B^{+}D_{1}+B^{+}C_{0}\{ (\tau-\nu)E_{I}^{+}0\Lambda_{r}(\tau-\nu)E^{\prime
and
\det C_{1}\neq 0
lC_{1}^{-1}D_{2}}0^{1}
\cdot
,
.
Thus
T. Ohkubo and T. Shirota
118
|q(x’, D’)
v_{l+1}^{+}|_{\frac{1}{2},f}\geq C\gamma|v_{l+1}^{+}|_{-\frac{1}{2},f}
with some C>O . Hence it holds for large
|B^{+}
.
\gamma
that
C_{0}v^{+}|_{1}2’ f\geq C(\gamma^{1}\Sigma|v_{I}^{+}|_{0,\gamma}+\gamma|v_{III}^{+}|_{-\frac{1}{2},\gamma})
.
On the other hand
|B^{+}
.
C_{0}v^{+}|_{\frac{1}{2},\gamma}\leq|C_{0}\cdot B^{+}v^{+}|_{\frac{1}{2},f}+|[B^{+}, C_{0}]v^{+}|_{\frac{1}{2},\gamma}
\leq C(|B^{+}v^{+}|_{\frac{1}{2},r},+|v^{+}|_{-_{\tau}^{1},\gamma})
.
Thus Theorem 5. 1 is proved.
structure of
II is not empty.
\S 6. The
(P_{1}, B_{1})
for the case where the set
and )
This section is devoted to the case where the conditions (II)
and its neighare applied. We only consider the point
for which the set II is not empty
borhood
. Note that in this case the boundary condition
and
may be replaced by
\gamma
\beta)
(x^{0}, \tau^{0}, \sigma^{0})\in\Gamma\cross R^{n}
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
(C_{-}\cross R^{n-1}))
(B_{1})
R(x^{0}, \tau^{0}, \sigma^{0})=O
(V_{I}^{+}, V_{I}^{-},\cdot V_{II}’, V_{II}’, V_{III}^{+}, V_{III}^{-})(x’, D’)
(B_{1})
.
t tuI+
(
, t VII,
u_{\acute{I}I}
,
, u VII,
u_{II}^{\prime\prime t}
tu_{III}^{-}
) (x’, O)=g(x’) ,
in
R^{n}
and it holds that
B^{+} (x’, \tau, \sigma)=(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})
,
R(x’, \tau, \sigma)=\det(V_{I}^{+}, V_{II}^{+}, V_{III}^{+})
,
(C_{-}\cross R^{n-1})) ,
analytic in
in
(C_{-}\cross R^{n-1})
(C-\cross R^{n-1})) .
and continuous in
in \S 3 of 2m\cross
We give some lemmas on the decomposed problem
2m system, which will be used in \S 8 for the proof of the estimate (1. 1).
For the functions defined in \S 3 and \S 4 let us also use the following
notations for simplicity in symbolic caluculations:
Hence
B^{+}
and R are
S_{+}^{0}
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
(\tau, \sigma)
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
\in U(\tau^{0}, \sigma^{0})\cap
(P_{1}, B_{1})
B’
(x’, \mu, \sigma)=(V_{I}^{+}, V_{II}’, V_{III}^{+}) (x’, \tau, \sigma)
,
B’
(x’, \mu, \sigma)=(V_{I:}^{+}V_{II}’, V_{III}^{+})(x’, \tau, \sigma)
,
and
(6. 1)
where
s_{21}(x, \mu, \sigma)=s_{1}(x, \mu, \sigma)+\sqrt{\mu}s_{2}(x, \mu, \sigma)
,
On structures
of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
117
Let
C_{I}(x’, \tau, \sigma)=D_{1}+C_{0} \{ (\tau-\nu)E_{I}^{+}0\Lambda_{\gamma}(\tau-\nu)E’0)C_{1}^{-1}D_{2}
.
\backslash
Then
C_{I}
is homogeneous of order 0 in
V_{I}^{-}
(x’, \tau, \sigma)=B^{+}\cdot C_{I}
in
(\tau, \sigma)
and
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
.
Thus the proof of the lemma is completed.
Now we must go back to the proof of Theorem 5. 1. Since
. Take
by (2. 6) as the extention and denote them by
we take
in the proof of Lemma 5. 3 (ii) and put
and
{\rm Im}\tau^{0}=O
(\tilde{\tau},\tilde{\sigma})
(\tilde{\tau}_{k},\tilde{\sigma}_{k})
C_{1}(x’, \tau, \sigma)
C_{0}(x’, \tau, \sigma)
\tilde{C}_{0}(x’, \tau, \sigma)=C_{0}(\tilde{x},\tilde{\tau},\tilde{\sigma})
\overline{C}_{1}(x’, \tau, \sigma)=C_{1}(\tilde{x},\tilde{\tau},\tilde{\sigma})=(V_{I}^{+}, a, V_{l+2}^{+}, \cdots, V_{m}^{+})
\tilde{C}_{2}(x’, \tau, \sigma)=\{\backslash
E_{I}^{+}0\tilde{\tau}-\nu(x, \tilde{\sigma})E0,)=(\backslash E_{I}^{+}0
(\tilde{x},\tilde{\tau},\tilde{\sigma})
q(x’, \tau, \sigma)E0
,)
1
Then we have with some C>O
(5. 5)
|\det
\overline{C}_{1}(x’, \tau, \sigma)|\geq C
for every
\cross\overline{C}_{-}\cross
(x’, \tau, \sigma)\in I
R^{n-1}
,
and it follows from (5. 3. 5) that
(5. 6)
\tilde{B}^{+}\tilde{C}_{0}(x’, \tau, \sigma)=\overline{C}_{1}\overline{C}_{2}(x’, \tau, \sigma)
Furthermore, we see from ( 2.
(5. 7)
.
and (5. 4) that
6)’
-{\rm Im}\tilde{q}(x’, \tau, \sigma)=\tilde{\gamma}+\nu_{2}(\tilde{x},\tilde{\sigma})\geq\tilde{\gamma}
=\tilde{Y}_{k}(\tau\Lambda_{\gamma}^{-1}, \sigma\Lambda_{f}^{1})\geq C\gamma\Lambda_{\gamma}^{1}
with some
C>O . Let
B^{+}
,
C^{0}
,
C_{1}
,
respectively.
it holds for large that
\tilde{B}^{+},\tilde{C}_{0},\tilde{C}_{1},\overline{C_{2}^{\cdot}},\tilde{q}\in S_{+}^{0}
\alpha)
and
\beta
) (ii)
, q be the operators with their symbol
Then from (5. 5), (5. 6) and Lemma 2. 1
C_{2}
\gamma
|B^{+}(x’, D’)
\cdot
C_{0}(x’, D’)v^{+}|_{\tau’ r}1
\geq C|C_{2}(x’, D’)v^{+}|_{\frac{1}{2},r}
\geq C
(
|v_{I}^{+}|_{\tau}1
,
r^{+}|Q\{x
, D’ )
v_{l+1}^{+}|_{\neq,r}+ \sum_{f=l+2}^{m}|v_{j}^{+}|_{\doteqdot,r})
If we consider the bilinear form -{\rm Im}
and Lemma 2. 1 ) (iv) and ) (iv) that for large
\langle Q\{x, D’) v, v\rangle_{0,\gamma}
\alpha
\beta
\gamma
.
, we see from (5. 7)
On structures of certain
s_{1}=(
L^{2}
-well-posed mixed problems for hyperbolic systems
-p_{11}\mu+\lambda_{1}(x, \mu, \sigma)-\lambda_{1}(x, O, \sigma)
=\mu s1(1)(x, \mu, \sigma)
)
119
(\Lambda_{0}^{(0)}+p_{12}\mu)^{-1}
,
s_{2}=\lambda_{2}(x, \mu, \sigma)(\Lambda_{0}^{(0)}+F_{12}\mu)^{-1}
and
, are real for real . (see Lemma 3. 1 and subsection 4. 1).
Here note that B’ and B’ are smooth and it holds that in U(x^{0})\cross(U (O,
s_{1}
s_{2}
\mu
\sigma^{0})\cap R^{n})
(6. 2)
|s_{1}^{(1)}
(x, \mu, \sigma)|\leq C,
and
s_{2}(x, \mu, \sigma)>0
with some C>0 . Here after we consider only , with
Then we have the following
LEMMA 6. 1.
For the fu nctions in \S 4 the relations (i), (ii) and (iii)
rectly from their defifinitions:
\tau
{\rm Im}\tau={\rm Im}\mu\leqq O
\mu
\alpha)
follow di-
(i)
h_{II}’=h_{II}^{+}-s_{21}h_{II}’
,
h_{II}^{-}=h_{II}^{+}+\alpha^{-1}(\lambda_{II}^{-}-\lambda_{II}^{+})h_{II}’
,
hence
V_{II}’=V_{II}^{+}-s_{21}V_{II}’
,
V_{II}^{-}=V_{II}^{+}+\alpha^{-1}(\lambda_{II}^{-}-\lambda_{II}^{+})V_{II}’
,
(ii)
b_{IIII}=R^{-1t}
\det B’=\alpha(\overline{b_{IIII}}-1)(\lambda_{II}^{-}-\lambda_{II}^{+})^{-1}
1-s_{21}b_{IIII}=R^{-1}
b_{ij}=\overline{b_{ij}}
.
\det B’
,
,
for every i and every
j\not\in II
,
(iii)
\det
\beta)
(
V_{I}^{+}
,
(1-s_{21}b_{IIII})V_{II}’-b_{II}
The condition (II)
\beta)
II
V_{II}’
,
V_{III}^{\dashv-})=O
.
implies
(i)
\det B’(x^{0}, O, \sigma^{0})=O
Now let
Q(x’, \mu, \sigma)\in S_{+}^{0}
be
and
\det B’(x^{0}, O, \sigma^{0})\neq 0
.
defifined as follows:
Q(x’, \mu, \sigma)=(\det B’)(\det B’)^{-1}
in
U(x^{0})\cross U
Thm it holds that
(ii)
Q(x’, \mu, \sigma)=(1-s_{21}b_{IIII}) (b_{IIII})^{-1}=R(\det B’)^{-1}-s_{21}
=
(\lambda_{II}^{-}-\lambda II)
(\alpha(b_{IIII}-1))^{-1}--s_{21}
.
(O,
\sigma^{0}
).
T. Ohkubo and T. Shirota
120
(iii)
\det
in
(V_{I}^{+}, QV_{II}’-V_{I1}’, V_{III}^{+})=O
U(x^{0})\cross U(0, \sigma^{0})
.
) (i) follows from the fact that (h_{II}^{+}, h_{II}’)=(h_{II}’, h_{II}’)S_{II} and
) (i) and ) (ii) implies ) (ii) and ) (iii) , re.
spectively. From ) and ) (i) it follows directly that ) (ii) and (iii) are
valid. Therefore we only prove ) (i) . From ) (i) and (6. 1) we have
PROOF.
\alpha
(h_{II}^{+}, h_{II}^{-})=(h_{II}’, h_{II}’)S_{II}T_{II}
\beta
\beta
\alpha
\alpha
\alpha
\alpha
\alpha
\beta
R(x^{0}, \tau^{0}-i\gamma, \sigma^{0})=
\det
\alpha
(V_{I}^{+}, V_{II}’+s_{21}V_{II}’, V_{III}^{+})
=
\det B’+s_{21}
=
(\det B’+s_{1} \det B’)
\det B’
(x^{0}, -i\gamma, \sigma^{0})
+\sqrt{-i\gamma}(s_{2}\det B’)(x^{0}, -i\gamma, \sigma^{0})
Hence it follows from
R(x^{0}, \tau^{0}, \sigma^{0})=O
.
that
(\det B’+s_{1}\det B’)
(x^{0}, O, \sigma^{0})=O
which implies \det B’(x^{0}, O, \sigma^{0})=O . Since (\det
analytic in , it follows from the condition (II)
B’+s_{1}\det B’)
\beta)
\mu
(s_{2}\cdot\det B’)(x^{0}, O, \sigma^{0})\neq 0
,
(x^{0}, \mu, \sigma^{0})
is
that
,
which together with (6. 2) implies ) (i) .
be
Lemma 6. 2. Assume the condition (II) . Let the equality
. Then there exist a constant C>O and operasatisfified for
independent of U such that for large
,
and
tors
\beta
(B_{1})
\beta)
U\in H_{\frac{1}{2},\gamma}(R_{+}^{n+1})
C_{1}
C_{2}\in S_{+}^{-1}
k_{JK}\in S_{+}^{0}
\gamma
|g-V_{III}^{-}u_{III}^{-}-C_{1}u_{I}^{-}-C_{2}u_{II}’|_{\frac{1}{2},r}
\geq C(|u_{I}^{+}+k_{II}u_{I}^{-}+k_{I}
II u_{II}’|_{1},f+2|u_{II}’+k_{III}u_{I}^{-}+Qu_{II}’|1 , f
+| u I+II +k_{III} I u_{I}^{-}+k_{III} II
Moreover, in
and has the form :
in
u_{II}’|_{2}1,)f
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap (\overline{C}_{-}\cross R^{n-1}))
, the symbol
of
(k_{JX})
is analytic
(\tau’, \sigma’)
(6. 3)
(\begin{array}{lllll}k_{II} k_{III} k_{IIII} \backslash k_{III} k_{II} k_{IIIII} k_{III} I k_{IIIII}II k_{IIIIII} \end{array})=(V_{I}^{+}, V_{II}’, V_{III}^{+})^{-1}(V_{I}^{-}jV_{II}’, V_{III}^{-})(x’,\tau’, \sigma’)
matrices with m-,l-1,1 or (m-l) for J=I, II
are
. or
III, respectively.
In the constant coefficients case the analogous inequality holds, which
follows directly from the proof of the above lemma:
COROLLARY 6. 1. Assume the same condition as in Lmma 6. 2.
where
k_{JK}
m_{J}\cross m_{K}
On structures
of certain
L^{2}
-well-posed mixed problems
hyperbolic systems
for
121
Let
(B_{1})_{x^{0}}
\hat{g}(\tau, \sigma)=(V_{I}^{+}, V_{II}’, V_{III}^{+})\cdot t(t\theta_{I}, \hat{u}_{II}^{\prime t},\theta_{III}^{+})
+ (Vt,
V_{II}’
,
V_{III}^{-}
)
\cdot t(tu_{I}^{-}, \theta_{II}’,{}^{t}\theta_{III}^{-})
Thm there exist a constant C>Oind\psi mdent
|
g-V_{III}^{-}\theta_{III}^{-}
+|\theta_{II}^{+}
OF
\theta
.
such that
I +k_{III} I
IIu^{\wedge}II’|+|\theta_{II}’+k_{II}
\theta_{I}^{-}+k_{III}
II
I
\theta_{I}^{-}+Q\theta_{II}’|
u_{II}^{\wedge}|’)
are fu nctions defifined by (6. 3).
LEMMA 6. 2. Define
by (6. 3). Then it holds that
U(\tau^{0}, \sigma^{0})\cap (\overline{C}_{-}\cross R^{n-1})
PROOF
of
(\tau, \sigma, O)
|
\geq C (|u_{I}^{\wedge+}+k_{II}u_{I}^{\wedge-}+k_{I}
in
(\tau, \sigma, O)
, where
(k_{JK})
k_{JK}
V_{I}^{-}
(x’, \tau, \sigma)=V_{I}^{+}k_{II}+V_{II}’k_{III}+V_{III}^{+}k_{IIII}
(6. 2. 1)
,
VI+II II
Here we used the relation k_{IIII}=Q . Hence we have from
V_{II}’(x’, \tau, \sigma)=V_{I}^{+}k_{III}+QV_{II}’+
k_{III}
(B_{1})
g(x’)=V_{I}^{+}(x’, D’)u_{I}^{+}(x’, O)+V_{II}’(x’, D’)u_{II}’(x’, O)
+
VI+II (x’, D’)u_{III}^{+}(x’, O)
+(V_{I}\cdot k_{II}+V_{II}’\cdot k_{III}+V_{III}^{+}\cdot k_{IIII}) u_{I}^{-}(x’, O)
+(V_{I}\cdot k_{III}+V_{II}’\cdot Q+V_{III}^{+}\cdot k_{IIIII})
+
u_{II}’(x’, O)
V_{III}^{-}(x’, D’)(u_{III}^{-}(x’, O)
+C_{1}(x’, D’)u_{I}^{-}(x’, O)+C_{2}(x’, D’)u_{II}’(x’, O)
where
C_{1}
and
C_{2}\in S_{+}^{-1}
,
arise from commutators. Hence we obtain
g-V_{III}^{-}u_{III}^{-}-C_{1}u_{I}^{-}-C_{2}u_{II}’
u_{I}^{+}+k_{I1}u_{I}^{-}+k_{I}
=(VI, V_{II}’, V_{III}^{+})
II
u_{\acute{I}1}
(x’, D’)\{u_{II}’+k_{II}1u_{I}^{-}+Qu_{I1}’
u_{III}^{+}+k_{III1}u_{I}^{-}+k_{III}
u_{II}’)^{1}\backslash
.
II
by Lemma 6. 1 ) (i) , the lemma follows from
Since
Lemma 2. 1 ) (ii) .
Lemma 6. 3. Consider the constant coeffiffifficients problem
of
)
system in a suffiffifficiently small neighborhood
:
\det\tilde{B}’(x^{0}, \tau^{0}, \sigma^{0})\neq O
\beta
\beta
(P_{II}, Q)_{x^{0}}
2\cross 2
(U(\tau^{0}, \sigma^{0})\cap (\Sigma
-
\prime P_{II}(x^{0}, \tau, \sigma, D_{n})\theta_{II}(\tau, \sigma, x_{n})
=(E_{II}D_{n}-M_{II}(x^{0}, \tau, \sigma)
(P_{II}, Q)_{x^{0}}
=\hat{f}_{II}(x_{n})
.
,
for
).
x_{n}>O
t(\theta\text{\’{I}}_{I}, \theta_{II}’)
,
Q(x^{0}, \tau, \sigma)\theta_{II}’(\tau, \sigma, O)+\theta_{II}’(\tau, \sigma, O)=O
.
(\tau, \sigma, x_{n})
T Ohkubo and T. Shirota
122
the conditions (II)
posed, that is, for any
-), the problm
satisfy ing
If
and (III) are assumed, then
and any
has a unique solution
\beta)
\hat{f}_{II}(x_{n})\in H_{1}(x_{n}>0)
(6. 4)
is
L^{2}
-well-
(\tau, \sigma)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
\theta_{II}(\tau, \sigma, x_{n})\in H_{1}(x_{n}>0)
(P_{II}, Q)_{x^{0}}
\Sigma
(P_{II}, Q)_{x^{0}}
||\theta II(\tau, \sigma, \cdot)||\leq C\gamma^{-1}||\hat{f}_{II}(\cdot)||
for sme
C=C(x^{0}, \tau^{0}, \sigma^{0})>0
.
PROOF. Recall the proof of Theorem 4. 1. First we obtain from the
. Transform
condition (III) the -well-posedness of the problem
it by a non-singular matrix
L^{2}
(P^{0}, B)_{x^{0}}
S(\tau, \sigma)=(h_{I}^{+}, h_{I}^{-}, h_{II}’, h_{II}’, h_{III}^{+}, h_{III}^{-})
.
Then we have the following constant coefficients problem equivalent to
(P^{0}, B)_{x^{0}}
:
]
O=F, for
(\tau, \sigma)
(P_{1}, B_{1})_{x^{0}}| \int_{B_{1}O=BSO}^{P_{1}O=[ED_{n}-}=0
=
(V_{I}^{+}, V_{II}’, V_{III}^{+}){}^{t}(\theta_{I}^{+},\hat{u}_{II}’,{}^{t}\theta_{III}^{+})
+ (VI-,
V_{II}’
,
V_{III}^{-}
)
x_{n}>0
,
(\tau, \sigma, 0)
{}^{t}(^{t}\theta_{I}^{-}, \theta_{II}’,{}^{t}\theta_{III}^{-})
(\tau, \sigma, 0)
,
where
O(\tau, \sigma, x_{n})=S^{-1}\theta(\tau, \sigma, x_{n})
=t({}^{t}\theta_{I}, {}^{t}\theta_{I}^{-},\hat{u}_{II}’,\hat{u}_{II}’,{}^{t}\theta_{III}^{+},{}^{t}\theta_{III}^{-})
and
F(x_{n})=S^{-1}\hat{f}(x_{n})
={}^{t}(^{t}\hat{f}_{I}^{+},{}^{t}\hat{f}_{I}^{-},\hat{f}_{II}’,\hat{f}_{II}’,,{}^{t}\acute{\grave{f}}_{III}^{+},{}^{t}\hat{f}_{III}^{-})
Existance of a solution of
for
solution O of
1)
(P_{1}, B_{1})_{x^{0}}
(P_{II}, Q)_{x^{0}}
. From Corollary 5. 1 the
\hat{F}(x_{n})={}^{t}(O, O,\hat{f}_{I}’,\hat{f}_{II}’, O, O)
\hat{u}_{I}^{-}(\tau, \sigma, x_{n})=\hat{u}_{III}^{-}(\tau, \sigma, x_{n})=O
.
Hence from Corollary 6. 1 it holds that
.\theta_{I}^{+}(\tau, \sigma, O)+k_{I}
(6. 3. 1)
II
\theta_{II}’(\tau, \sigma, 0)=O
\theta_{II}’(\tau, \sigma, O)+Q\theta_{II}^{J}(\tau, \sigma, O)=O
\theta_{III}^{+}(\tau, \sigma, O)+k_{III}
II
.
,
,
\theta_{II}’(\tau, \sigma, O)=O
.
fufills
On structures
Therefore the
{}^{t}(\hat{f}_{II}’,\hat{f}_{II}’)
of certain
t(u\text{\’{I}}_{I}, \hat{u}_{II}’)
L^{2}
-well-posed mixed problems for hyperbolic systems
(\tau, \sigma, x_{n})
is a solution of
(P_{II}, Q)_{x^{0}}
for
123
\hat{f}_{II}(x_{n})=
.
The estimate (6. 4) for the above solution follows from the
posedness of
:
2)
L^{2}
-well-
(P_{1}, B_{1})_{x^{0}}
||
O(\tau, \sigma, \cdot)||\leq C\gamma^{-1}||F(x_{n})||
Uniqueness of the solution of
corresponding to
solution of
of the equations
3)
(P_{II}, Q)_{x^{0}}
. Let
be the
. Take the bounded solutions
(P_{II}, Q^{t})_{x^{0}}
\hat{f}_{II}=O
.
\hat{u}_{II}={}^{t}(\theta_{II}’,\hat{u}_{II}’)
\hat{u}_{I}^{+},\hat{u}_{III}^{+}
\{
(
E_{I}^{+}D_{n}-\lambda_{I}^{+} (\tau, \sigma)
)
\theta_{I}^{+}
(\tau, \sigma, x_{n})=O
for
x_{n}>0
for
x_{n}>0,\cdot
,
,
4_{I}^{+}(\tau, \sigma, O)=-k_{III}\theta_{II}’(\tau, \sigma, 0)
and
\{
(
E_{III}^{+}D_{n}-M_{III}^{+}(\tau, \sigma)
)
\theta_{III}^{+}(\tau, \sigma, O)=-k_{III}
\hat{u}_{III}^{+}(\tau, \sigma, x_{n})=O
II
\theta_{II}’(\tau, \sigma, 0)
.
Put
O={}^{t}(^{t}\theta_{I}^{+}, O, \theta_{II}’, \theta_{II}^{\prime\prime t},\theta_{III}^{+}, O)
Then
P_{1}O=O
.
and (6. 3. 1) is fulfilled. Hence it follows from (6. 2. 1) that
B_{1}O=(V_{I}^{+}, V_{II}’, V_{III}^{+})\cdot
=(VI, V_{II}’, V_{III}^{+})
t(t\theta_{I}, \theta_{II}’,{}^{t}\thetaIII+)
+V_{II}
\cdot(\begin{array}{lll}\theta_{I}^{+}+k_{I} II \hat{u}_{II}’\theta_{II}’,+Q\theta_{I},I \theta_{III}^{+}+k_{III} IIu’II\end{array})=O
\theta_{II}’
.
with \hat{F}=O . From the -wellTherefore O i s a solution of
. This proves
posedness of
we obtain O=O and hence
the lemma.
Lemma 6. 4. Assume the conditions (II)
and ).
(i) Then the fu nction Q(x’, \mu, \sigma) defifined in Lmma 6. 1 ) (i) takes
real values in U(x^{0})\cross U (O, ) for real .
(ii) Suppose, in addition, the condition (III). Then for
L^{2}
(P_{1}, B_{1})_{x^{0}}
\theta_{II}’=\theta_{II}’=O
(P_{1}, B_{1})_{x^{0}}
\beta)
\gamma
\beta
\sigma^{0}
\mu
(x’, \tau, \sigma)\in U(x^{0})
\cross U(\tau^{0}, \sigma^{0})
O,
,
R(x’, \tau, \sigma)=O
for
{\rm Im}\tau\leq O
is equivalent to
\tau=\theta(x’, \sigma)
and Q(x’ ,
\sigma)=O.
PROOF.
Since, in the case (a), the choice of the vectors
stricted to be real for \mu\geq 0 , Lemma 4. 1 and the condition (II)
(i)
that
\{h_{I}^{f}, h_{II}^{\pm}\}
\gamma)
are reimply
T. Ohkubo and T. Shirota
124
b_{IIII}^{-}(x’, \tau, \sigma)
=
\det
B(h_{I}^{+}, h_{II}^{-}, h_{III}^{+})\cdot
(\det B(h_{I}^{+}, h_{II}^{+}, h_{III}^{+}))^{-1}
is real for \mu\geq 0 and R(x’, \tau, \sigma)\neq 0 . Here note that
which were defined in \S 4 by
h_{II}^{\pm}
are the eigenvectors
(h_{II}^{+}, h_{II}^{-})=(h_{II}’, h_{II}’)S_{II}T_{II}
and hence are real for
\mu\leq 0 that
\mu\geq 0
. From Lemma 6. 1
\beta
) (ii) it holds for
{\rm Im}
Q(x’, \mu, \sigma)=R(\det B’)^{-1}-s_{21}
(6_{\tau}.4.1)
=
(\lambda_{II}^{-}-\lambda_{II}^{+})
(\alpha(b_{IIII}-1))^{-1}--s_{21}
.
is real
If R(x’, \tau, \sigma)=O and \mu\geq O then Q is real at such points, because
for \mu\geq 0 by virture of (6. 1). If R(x’, \tau, \sigma)\neq 0 , \mu\geq 0 and b_{IIII}^{-}\neq 1Q is also
, ,
real, because
are so. From Lemma 6. 1 ) (ii) we have
II and
S_{21}
\overline{b_{II}}
\lambda_{II}^{\pm}
\tilde{b_{IIII}}(x^{0},
\alpha
S_{21}
\alpha
\tau^{0}-i\gamma
,
\sigma^{0}\rangle-1
=-2\sqrt{-i\gamma}(\lambda_{2}\cdot\det B’\cdot\alpha^{-1})
(x^{0}, \tau^{0}-i\gamma, \sigma^{0})
.
R^{-1}
( ,
x^{0}
\tau^{0}-
-ir,
\sigma^{0}
)
\iota
and
the condition (II)
Since
. Thus we see that Q(x’, \mu, \sigma) tqkes real values
imply
II
. Since Q is analytic in at \mu=O Q always takes real
whenever
values for real .
(ii) Suppose
and Q(x’, O, \sigma)=0. Then from (6. 1) and
(6. 4. 1) we have R(x’, \theta(x’, \sigma), \sigma)=O . Hence we have only to prove the
, then from Theorem
converse. Suppose that R (x’, \tau, \sigma)=O and
\mu=O . First if \mu<0 , it follows from (6. 1) that
4. 1
and from (i) that Q is real. This contradicts (6. 4. 1). We shall secondly
show that R(x’\tau, \sigma)=0 for \mu>0 is also impossible under the condition
. From
(III). Let
be a Lopatinskii determinant of
Lemma 6. 3 and Theorem 3 of [11] we easily see the condition (III) implies
that for every fixed x’\in U(x^{0})
(\lambda_{2}\cdot\det B’\cdot\alpha^{-1}) (x^{\phi}, \tau^{0}, \sigma^{0})\neq 0
\beta)
R(x^{0}, \tau^{0}, \sigma^{0})=O
(x^{0}, \check{\tau}^{0}, \sigma^{0})\neq 1
\tilde{b_{II}}
\mu\not\geq O
\mu
\mu
\tau=\theta(x’, \sigma)
{\rm Im}\tau\leq 0
{\rm Im}\tau=
{\rm Im} s_{21}\neq 0
{\rm Im}
(P_{II}, Q)_{x^{0}}
R_{II}(x^{0}, \tau, \sigma)
(6. 4. 2)
R_{II}(x’, \tau, \sigma)\neq O
for
\tau-\theta(x’, \tau)
>O
(for the case (a) in Lemma 3. 1.)
On the other hand we see in \S 4 that
M_{II}S_{II}=(\begin{array}{ll}\lambda_{II}^{+} \alpha 0 \lambda_{II}^{-}\end{array})
Hence letting
S_{II}^{1}
be the first column of
S_{II}
we obtai n
On structures of certain
L^{2}
125
-well-posed mixed problems for hyperbolic systems
R_{II}(x’, \tau, \sigma)=(Q, 1
(6. 4. 3)
)
S_{II}^{1}
=Q +s_{21}=R(\det B’)^{-1}
where the last equality follows from (6. 4. 1). Thus (6. 4. 2) and (6. 4. 3)
imply that
R(x’, \tau, \sigma)\neq 0
for \mu=\tau-\theta(xf, \sigma)>0
from which we obtain
If
\mu=O
and
\mu=O .
Q(x’, O, \sigma)\neq 0
R ( x’ ,
\theta(xf, \sigma)
,
\sigma)=
then we have from (6. 1) and (6. 4. 1)
Q(x’, O, \sigma)(\det B’)(x’, O, \sigma)\neq O
.
This is a contradiction, hence Q(x’, O, \sigma)=O holds. Thus the proof of (ii)
is completed.
LEMMA 6. 5. Assume the conditions (II) ,
and {III). Then there
exists a neighborhood
such that for every
\gamma)
\beta)
(x’, \sigma)\in U(x^{0}, \sigma^{0})
U(x^{0}, \sigma^{0})
-Q(x’, O, \sigma)\geq 0
COROLLARY 6. 2.
have
gr,ad(x,\sigma)
,
in the case (a).
Under the same conditions as in Lemma 6. 5 we
Q(x’, O, \sigma)=O
on
\{(x’, \sigma)\in
R^{2n-1}
;
Q(x’, O, \sigma)=O\}
PROOF OF Lemma 6. 5. The condition (III) together with Lemma 6. 3
implies (6. 4. 2). On the other hand we have from (6. 4. 3) and (6. 1)
R_{II} (x’, \tau, \sigma)=Q+\mu s_{1}^{(1)}+\sqrt{\mu}s_{2}
Regarding our convention
Then we see
(6. 5. 1)
\sqrt{1}=-1
.
we consider the equation
R_{II}=O
.
\sqrt{\mu}S_{2}(x’, 0, \sigma)=-Q(x’, 0, \sigma)+r(x’, \sqrt{\mu}, \sigma)\mu
is an analytic function in
where
such that
exists a sequence
r(x’, \sqrt{\mu}, \sigma)
\sqrt{\mu}
. Now
suppose that there
(x^{n}, \sigma^{n})
(x^{n}, \sigma^{n})
-
(x^{0}, \sigma^{0})
,
as
narrow\infty,\cdot
-Q(x^{n}, O, \sigma^{n})<0
with \mu^{n}>0 such that (6. 5. 1) holds for
Then there exists a sequence
R(x^{0}, O, \sigma^{0})=O
, because
from (6. 2) and
we have a sequence
from Lemma 6. 1 ) (ii) . Putting
such that
\mu^{n}>0
.
and
\{\sqrt{\mu^{n}}\}
(x^{n}, \mu^{n}, \sigma^{n})
s_{2}(x^{n}, O, \sigma^{n})>0
\beta
Q(x^{0}, O, \sigma^{0})=
\tau^{n}=\mu^{n}+\theta(xn, \sigma^{n})
\{\tau^{n}\}
R_{II}(x^{n}, \tau^{n}, \sigma^{n})=O
T. Ohkubo and T. Shirota
126
This contradicts (6. 4. 2) and proves the lemma.
PROOF 0F COROLLARY 6. 2. Assume that
gr” ad(x\sigma)
Q(x^{0}, O, \sigma^{0})\neq 0
and for simplicity let
\frac{\partial Q}{\partial x_{0}}(x^{0}, O, \sigma^{0})\neq 0
t
Then from the implicit function theorem we have
Q(x’, O, \sigma)=(x_{0}-c_{1}(x’, \sigma)
)
c_{2}(x’, \sigma)
and , are real, because Q(x’, O ,
in (x’, \sigma) ,
where is
.
is real. This contradicts Lemma 6. 5, hence
such that
Furthermore the same argument is valid for the points
Q(x’, O, \sigma)=0.
Hence the corollary is proved.
REMARK 6. 1. In the case (b) in Lemma 3. 1 the discussions of Lemma
6. 4 and 6. 5 are also valid. In this case the conclusion of Lemma 6. 5
becomes -Q(x, O, \sigma)\leq 0 .
REMARK 6. 2. If the coefficients of B are real and the set III is empty,
is fulfilled automatically. Because it then holds that
the condition (II)
C^{\infty}
c_{2}(x^{0}, \sigma^{0})\neq 0
c_{1}
c_{1}
C_{2}
\sigma)
gr” ad(x\sigma)Q(x^{0}, O, \sigma^{0})=O
(x’, \sigma)
\gamma)
\tilde{b_{IIII}}=\det B(h_{I}^{+}, h_{II}^{-})(\det B(h_{I}^{+}, h_{II}^{+}))^{-1}
LEMMA 6. 6.
Assume the conditions (II)
and (III).
\beta)
Then it holds
that
(i)
(ii)
and
V_{I}^{-}(x’, \tau, \sigma)\in L(V_{I}^{+}, V_{III}^{+})
V_{II}’(x’, \tau, \sigma)\in L(V_{III}^{+})
for such points
O,
.
(x’, \eta, \sigma)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap R^{n})
that
\eta=\theta(x’, \sigma)
and Q(x’ ,
\sigma)=O
be the fixed real point satisfying the conditions
PROOF. Let
in Lemma 6. 6.
(i) Then from Theorem 4. 1 it holds that for some C(x’, \tau^{0}, \sigma^{0})>O
(x’, \eta, \sigma)
|\det
(6. 6. 1)
(V_{1}^{+}, V_{k}^{-}, V_{III}^{+})(x’, \eta-i\gamma, \sigma)|
\leq C(x’, \tau^{0}, \sigma^{0})|{\rm Im} \lambda_{II}^{+}(x’, \eta-i\gamma, \sigma)|^{*}
.
for
k\in I_{-}
. From Lemma 6. 1
|R(x’, \eta-i\gamma, \sigma)|
\beta
.
\gamma^{-z}1
) (ii) and (6. 1) we have
R(x’, \tau, \sigma)=(Q
+\sqrt{\mu}(\sqrt{\mu}s_{1}^{(1)}+s_{2}))\det B’
On structures
Since
of certain
Q(x’, O, \sigma)=0
L^{2}
-well-posed mixed problems for hyperbolic systems
127
we see that
Q(x’, \mu, \sigma)=\mu c_{0}(x’, \mu, \sigma)
with some function.
c_{0}
. Hence we have for such a
point
R(x’, \tau, \sigma)=\sqrt{\mu}(\sqrt{\mu}c_{0}+\sqrt{\mu}s_{1}^{(1)}+s_{2})\cdot\det B’
,
Hence
(6. 6. 2)
|R(x’, \tau, \sigma)|\leq C|\tau-\theta(xf, \sigma)|^{1}z=C\gamma^{1}2
where C>0 and
\tau=\eta-i\gamma
(6. 6^{\theta}.3)
,
. On the other hand
{\rm Im}\lambda_{II}^{+}(x’, \eta-i\gamma, \sigma)\leq C\gamma^{\frac{1}{2}}
with some C>0 . Hence it follows from (6. 6. 1), (6. 6. 2) and (6. 6. 3) that
\det(V_{I}^{+}, V_{k}^{-}, V_{III}^{+})(x’, \eta, \sigma)=O,\cdot
k\in I_{-}
,
which implies our assersion (i).
(ii) From Theorem 4. 1 it holds that
|\det
(V_{I}^{+}, \cdots,\check{V}_{I}’f,, \cdots, V_{l-1}^{+}, V_{I1}’, V_{III}^{+})(x’, \eta-i\gamma, \sigma)|
\leq
C\{x’ ,
.
for
j\in I_{+}
j\in I_{+}
.
,
\sigma^{0}
)
|{\rm Im}\lambda_{II}^{-}(x’, \eta-i\gamma, \sigma)|^{\not\in}
|R(x’, \eta-i\gamma, \sigma)|
. Hence the same
\det
for
\tau^{0}
argument
.
\gamma^{-\not\equiv}
,
as in (i) leads us to
(V_{I}^{+}, \cdots,\check{V}_{II}’f, \cdots, V_{l-1}^{+}, V_{II}’, V_{III}^{+})(x’, \eta, \sigma)=O
On the other hand for such points
\det
(x’, \eta, \sigma)
we have
(V_{1}^{+}, \cdots, \tilde{V}_{i}^{+}j, \cdots, V_{l-1}^{+}, V_{II}’, V_{III}^{+})=O
, because R(x’, \eta, \sigma)=O . Furthermore
and
, because of
are linearly independent in
\neq 0 .
Hence we obtain for any vectors
for
j\in I_{+}
i\in I_{+}\cup III_{+}
V_{III}^{+}\}
\{V_{I}^{+}
,
V_{II}’
,
\det B’(x^{0}, \tau^{O}, \sigma^{0})
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\{v_{i}\}\subset R^{n}
\det
(v_{1}, \cdots, v_{l-1}, V_{II}’, V_{III}^{+})=O
This implies that for such points
V_{II}’\in L(V_{III}^{+})
by virture of the linear independence of
,
\{V_{III}^{+}\}
the lemma.
LEMMA 6. 7. Assume the conditions (II)
Thm
in (6. 3) are functions in
and
k_{III}
k_{III}
and completes the proof of
\beta)
,
\gamma)
and (III).
T. Ohkubo and T. Shirota
128
C^{\infty}
) and analytic in
(x’,
\sigma)
belonging to some
every
( U(x^{0})\cross U (O,
and for
(\mu, \sigma)
\sigma^{0})
|kII1(x’, O, \sigma)|^{2}
({\rm Im}\mu\leq 0)
, and
neighborhood
for some C>0
U(x^{0}, \sigma^{0})
,
and
(6. 5)
|k_{1}
II(x’, O, \sigma)|^{2}\leq C|Q\{x’
,
0, ) .
\sigma
|
From (6. 2. 1) and Lemma 6. 6 we have for every
such that \mu=\tau-\theta(xf, \sigma)=O and Q(x’, O, \sigma)=O
Proof.
k_{I}
II
Hence it bolds for every
(6. 7. 1)
k_{I}
(x’, \mu, \sigma)=
(x’, \sigma)
II
kII 1 (x’. \mu, \sigma)=0
satisfying
1
Q(x’, O, \sigma)=0
that
(x’, O, \sigma)=k_{II1}(x’, O, \sigma)=Ot
On the other hand, from Lemma 6. 1
R (x’ ,
(x’, \mu, \sigma)
\theta(xf, \sigma)
,
\sigma)=
\beta
) we
have that for
.
Q(x’, O, \sigma) \det B’(x’, O, \sigma)
(x’, \sigma)\in U(x^{0}, \sigma^{0})
.
Therefore from the condition (II) , Lemma 6. 5 and its corollary, we see
of
at
that there exists a coordinate system { , , , , ,
such that
\beta)
y_{1}
\cdots
y_{N}
z_{1}
\cdots
z_{2n-1-N}\rangle
R^{2n- 1}
(x^{0}, \sigma^{\Gamma_{1}})
Q(x’, O, \sigma)=F(Y, Z)
= \sum_{i,f=1}^{N}\frac{\partial^{2}F(O,Z)}{\partial y_{i}\partial y_{j}}y_{i}y_{j}+G(Y, Z)
where
Y=Y(x’, \sigma)=
Z(x^{0}, \sigma^{0})=O
{yl9
\cdots
,
y_{N}
),
Z=Z(x’, \sigma)=(z_{1}, \cdots, z_{2n-1-N}) ,
Y(x^{0}, \sigma^{0})=O,
,
\frac{\partial^{2}F(O,Z)}{\partial y_{i}\partial y_{J}}<0
and
|G(Y, Z)|\leq C|Y|^{3}
with some C>0 .
Theorefore Q(x’, O, \sigma)=O on Y=O, where (6. 7. 1) is valid.
Furthermore from the above form of Q(x’, O, \sigma) we see that it satisfies
(6. 5).
Thus we see our assertions.
\S 7. Problems for
2\cross 2
system
of first order.
This section is devoted to problems for
system which play an
essential role for the proof of the estimate (1. 1), in the case where the set
II is not empty. In subsection 7. 1 apriori estimates for the decomposed
problem
in \S 3 are given by the modification of Kreiss’ consideratio
2\cross 2
(P_{II})
On structures
of certain
L^{2}
-well-posed mixed proble ms for hyperbolic systems
129
used for the opshould be noted that the symmetrizer
is conin a neighborhood
of the point
structed along the ‘surface’
. In subsection 7. 2 we consider
a boundary value problem (P_{II}, Q) for
system, whose main purpose
is to see how the essential part of the proof of (1. 1) is analyzed. It will
,
also be seen why the function Q was considered in \S 6. The point
,
considered in this section is only the one where the set II is not
).
empty (and hence
7. 1. Let
be the matrix defined in Lemma 3. 2 and as in
[7]. It
erator
R_{0}(x, \tau, \sigma)
P_{II}
(x^{0}, \tau^{0}, \sigma^{0})
U(x^{0})\cross U(\tau^{0}, \sigma^{0})
\tau=\theta(x, \sigma)
2\cross 2
(x^{0}
\tau^{0}
\sigma^{0})
\sigma^{0}\neq O
M_{II}(x, \mu, \sigma)
\S 6
put
\tau=\eta-i\gamma
,
and
\mu=\tau-\theta(x, \sigma)=e- ir
\Lambda_{\gamma}=
Expand
(|\tau|^{2}+|\sigma|^{2})
-.
in
around the surface
Then it is seen from Lemma 3. 2 that
M_{II}
U(x^{0})\cross U(O, \sigma^{0})
\tau=\theta(x, \sigma)
, i.e ., \mu=O .
M_{II}(x, \mu, \sigma)
=
(7. 1)
Mu{x,
\epsilon,
\sigma
) +M_{II}(x, \epsilon-i\gamma, \sigma)-M_{II}(x, \epsilon, \sigma)
=M_{II}(x, \tau,\sigma)+\epsilon E(x,\epsilon, \sigma)+(-i\gamma)H(x, \epsilon, \sigma)+O(r^{2}\Lambda_{f}^{1})
where E, H are real valued
functions\in S_{+}^{0}
’
\overline{M_{II}}(x, \tau, \sigma)=(^{\lambda_{1}(x_{O}O,\sigma)}
E (x,
\epsilon,
\sigma
)
=(\begin{array}{ll}e_{11} e_{12}^{\backslash }e_{2}.1 e_{22}\end{array})
,
.-
in
(\epsilon, \sigma)
\lambda_{1}(x,O\Lambda_{r}, \sigma)
,
and
),
H(x, \epsilon, \sigma)=(\begin{array}{ll}h_{11} h_{12}h_{21} h_{22}\end{array})
and consider the characteristic polynomial
\det
(
E_{II}\lambda-M_{II}(x, \mu, \sigma))=(\lambda-\lambda_{I1}^{+}(x, \mu, \sigma)
we see from putting
\lambda=\lambda_{1}(x, O, \sigma)
(\Lambda_{f}e_{21})(x, O, \sigma)=
(7. 2)
)
(\lambda-\lambda_{II}^{-}(x, \mu, \sigma)
),
in the above equation that
(\Lambda_{f}h_{21})
(x, O, \sigma)=\lambda_{2}(x, O, \sigma)^{2}>O
in the case (a),
(\Lambda_{r}h_{21})
(x, O, \sigma)=-\lambda_{2}’(x, O, \sigma)^{2}<O
in the case (b)
or
(\Lambda_{r}e_{21})(x, O, \sigma)=
with the notations in Lemma 3. 1.
As in [7] let us define in some
of order 0 by
symmetric
matrix
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap (\overline{C}_{-}\cross R^{n-1}))
2\cross 2
R_{0}
R_{0}(x, \tau, \sigma)=i
(D (x,
\tau,
\sigma
) -i\gamma’
F+
eE\{x, ,
\tau
\sigma
)).
a skew
T. Ohkubo and T. Shirota
130
where
D(x, \tau, \sigma)=\{\begin{array}{ll}O d_{1}d1 d_{2}(x,\tau,\sigma)\end{array}\}
,
F=(\begin{array}{ll}O -ff O\end{array})
d_{1}
Choose
b
,
d_{2},f
and
b
B(x, \tau, \sigma)=(^{b(x_{0}\tau,\sigma)}’
are real and in
(D+\epsilon’B)(C+\epsilon’E)
,
.
is a self-adjoint matrix,
(\tau, \sigma)\in\overline{\Sigma_{-}}
(7. 4)
b(x, \tau, \sigma)=(1+\epsilon e_{12})^{-1}(d_{1}e_{11}-d_{1}e_{22}+d_{2}e_{21})
Let
Then we obtain
and E is the matrix defined above.
C=(\begin{array}{ll}O 1O O\end{array})
that for
S_{+}^{0}
oO)
so that
(7. 3)
where
,
(\tilde{\tau},\tilde{\sigma})
be one of
(\tilde{\tau}_{k},\tilde{\sigma}_{k})
.
defined by (2. 6) and put
\tilde{R}_{0}(x, \tau, \sigma)=R_{0}(\tilde{x}(x),\tilde{\tau}(\tau\Lambda_{\gamma}^{-1}, \sigma\Lambda_{\gamma}^{-1})
,
\tilde{\sigma}(\tau\Lambda_{\overline{r}}^{1}, \sigma\Lambda_{\gamma}^{-1}))\in S_{+}^{0}
Hereafter, in this section we consider only such functions
where
and
with
. Then we have the following
so that there
LEMMA 7. 1. ([7]). One can choose
and u(x)
such that for every
exist constants C, C’ ,
H_{1.\gamma}(R_{+}^{n+1})
that
u(x)={}^{t}(u_{1}, u_{2})\in
u=\psi(D’)\cdot
t(v_{1}, v_{2})
,
t(v_{1}, v_{2}) \in H_{1,\gamma}(R_{+}^{n+1})
\psi(\tau, \sigma)=
supp\psi’\subset S_{2*_{0}}(\tau^{0}, \sigma^{0})
\psi’(\tau\Lambda_{\overline{r}}^{1}, \sigma\Lambda_{\gamma}^{-1})\in S_{+}^{0}
R_{0}(i.e., d_{1}, d_{2}, f)\in S_{+}^{0}
\gamma_{0}>0
(P_{II})
\gamma\geq\gamma_{0}
||P_{II}(x, D)u||_{0,\gamma}^{2}\geq C(T^{2}(1\cdot||u1||_{0,r}^{2}+C’||u_{2}||_{0,r}^{2})
+\gamma {\rm Im}
\langle R_{0}(x’, D’)u, u\rangle_{0,\gamma}),\cdot
where C’ can be taken however large
and
symbol of R_{0}(x, D’) is
\tilde{R}_{0}(x, \tau, \sigma)
{\rm Im}
if
f is taken large enough, and the
\langle R_{0}(x’, D’)u, u\rangle_{0,\gamma}
=2
{\rm Re}
(d2u2,
-2\gamma{\rm Im}
u_{1}\rangle_{0,\gamma}+{\rm Re}
\langle d_{2}u2, u_{2}\rangle_{0,\gamma}
\langle f\Lambda^{-1}u_{2}, u_{1}\rangle_{0,\gamma}+{\rm Re}
\langle(\epsilon b)u_{1}, u_{1}\rangle_{0,\gamma}
.
In order to avoid the ambiguity in the use of this lemma in what
follows, we sketch the
PROOF. Estimating the commutator, we have
2
{\rm Re}(P_{II}u, R_{0}u)_{0,\gamma}
\geq {\rm Im}
\langle R_{0}(x’, 1\mathcal{Y})u, u\rangle_{0,\gamma}+(((R_{0}M_{II})+(R_{0}M_{II})^{*})u,
-C(||u||_{0,\gamma}+|u|_{-\frac{1}{2},\gamma})
u)_{0,\gamma}
On structures of certain
L^{2}
-well-posed mixed problems
with C>0 independent of .
The symbol of
\gamma
R_{0}
(
x\sim(x),\tilde{\tau}
,
\tilde{\sigma}
(
-M_{II}^{*}
)
M_{II}(\tilde{x}(x),\tilde{\tau}
x\sim(x),\tilde{\tau}
,
\tilde{\sigma}
)
,
\tilde{\sigma}
).
for
hyperbolic systems
((R_{0}M_{II})+(R_{0}M_{II})^{*})(x, D’)
is
\Lambda_{\gamma}
R_{0}(\tilde{x}(x),\tilde{\tau}
,
\tilde{\sigma}
). .
\Lambda_{r}
Using (7. 1) and (7. 3) to calculate the symbol, we obtain for any
) ) and for any constant vector
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap
131
(x, \tau’, \sigma’)\in
u={}^{t}(u_{1}, u_{2})
(\overline{\Sigma}
-
u,
\backslash /(R_{0}M_{II}-M_{II}^{*}R()) (x,\cdot\tau’, \sigma’)
\geq-
C_{1}\gamma
+\gamma
’
’ (\epsilon’+\gamma’)
|u|^{2}
{\rm Re}\{\langle d_{1}h_{21}u_{1}, u_{1}\rangle+\langle d_{1}h_{22}u_{2}, u_{1}\rangle
+\backslash /(d_{1}h_{11}+ d_{2}h_{21}) u_{1}
where
u_{/}^{\backslash }
,
u_{2/}\backslash +(\backslash d_{1}/h_{12}+d_{2}h_{22}+f) u_{2}
does not depend on { x,
we have
C_{1}=C_{1}(D, F)>0
\tau’,
\sigma’)
,
u_{2}
/}
. For any
\delta>0
and
(x, \tau’, \sigma’)\in U(x^{0})\cross U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}})
|\langle d_{1}h_{22}u_{2}, u_{1}\rangle
+\langle d_{1}h_{11}+d_{2}h_{21})u_{1}, u_{2}\rangle|
\leq C_{2}(\delta|u_{1}|^{2}+\delta^{-1}|u_{2}|^{2})
with some C_{2}=C_{2}(D)>0 . Choose
and f=const such that for
,
sufficiently small, and
U(\tau^{0}, \sigma^{0})
d_{1}=const
(x, \tau’, \sigma’)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}}
(7. 1. 1)
d_{1}h_{21}(x, \epsilon’, \sigma’)\geq 2
(7. 1. 2)
d_{1}h_{12}+d_{2}h_{22}+f-\delta^{-1}C_{2},>Ot
(We see from (7. 1. 2) that we can
large enough.) Since
(\tilde{x},\tilde{\tau},\tilde{\sigma})\in
,
choose
d_{2}
arbitrarily if choosing f>0
we see from ( 2. 6)’ that
S_{-\epsilon_{0}}(x^{0})\cross S_{8*_{0}}(\tau^{0}, \sigma^{0})
for every
(x, \tau, \sigma)\in R_{+}^{n+1}\cross\overline{C_{-}}\cross R^{n-1}
\backslash /(R_{0}M_{II}-M_{II}^{*}R_{0})
(\tilde{x},\tilde{\tau},\tilde{\sigma})
\geq-\tilde{7}\Lambda_{\gamma}(\tilde{\epsilon}+\tilde{\gamma})
\Lambda_{\gamma}u
,
u_{/}^{\backslash }
C_{1}|u|^{2}+ \tilde{\gamma}\Lambda_{\gamma}(\frac{3}{2}|u_{1}|^{2}+(C’ +1)
\geq(_{\frac{3}{2}}-12\epsilon_{0}C_{1})\tilde{7}\Lambda_{\gamma}|u1|^{2}+(C’+1-12\epsilon_{0}C_{1})
|u_{2}|^{2})
\tilde{\gamma}\Lambda_{r}|u_{2}|^{2}
,
where C’>O is a constant such that
C’+1
(7. 1. 3)
Since
.
supp\psi’\subset S_{2- 0}
\geq d_{1}h_{11}+d_{2}h_{22}+f-\delta^{-1}C_{2}
it follows from ( 2.
) (v) that for large
(\tau^{0}, \sigma^{0})
see from Lemma 2. 1
\beta
\gamma
6)’
that
.
\tilde{\mathcal{T}}\Lambda_{\gamma}\psi’=\mathcal{T}\psi’
. Hence we
T. Ohkubo and T. Shirota
132
{\rm Re}
(
((R_{0}M_{II})+(R_{0}M_{II})^{*})(x, D’)u
\geq\gamma(1\cdot||u1||_{0,r}^{2}+C’
||u_{2}||_{0,\gamma}^{2}
, u)
,v
)
with some C’>0 . Hence from (7. 10) we obtain
(\delta\gamma)^{-1}||P_{II}u||_{0,\gamma}^{2}+\delta\gamma||u||_{0,\gamma}^{2}\geq\gamma(1\cdot||u1||_{0}^{2}
+
{\rm Im}
\backslash /
Q{x’, D’ ) u ,
u/0 ,
, \gamma+C’||u2||_{0}^{2} ,
\gamma
)
.
\gamma
Choosing \delta>O small we complete the proof.
In the above proof, choose
with , satisfying (7. 1. 1), (7. 1. 2)
and d_{2}=const>0 sufficiently large. Then we obtain the following
COROLLARY 7. 1.
(i) There exists a constant
and C_{1}>0
such that for every
there exist constants , C_{3}>0 satisfy ing
R_{0}
d_{1}
f
\gamma\geq\gamma_{0}
\gamma_{0}
C_{2}
(P_{II})’
for every
C_{2}||P_{II}u||_{0,\gamma}^{2}+C_{1}\gamma|u_{1}|_{0,\gamma}^{2}\geq C_{3}\mathcal{T}^{2}||u||_{0}^{2}
,
\gamma+1\cdot\gamma|u_{2}|_{0}^{2},\cdot
u(x) .
and a neighborhood
There exist a constant
,
, C_{1}>0 and
for every
there exist constants , C_{3}>O satisfy ing
(ii)
\gamma_{0}
\gamma\geq\gamma_{0}
U(\tau^{0}, \sigma^{0})
\^u=
(\tau\Lambda_{\gamma}^{-1}, \sigma\Lambda_{\gamma}^{-1})\in U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}}
such that
{}^{t}(\theta_{1}, \theta_{2})(x_{n})\in
C_{2}
H_{1}(R_{+}^{1})
C_{2}||P_{II}(x^{0}, \tau, \sigma, D_{n})
(P_{II})_{x^{0}}’
\geq C_{3}\gamma^{2}||
\^u
\^u
(\cdot)||^{2}+C_{1}\gamma|\theta_{1}|^{2}
(\cdot)||^{2}+1\cdot\gamma|\theta_{2}|^{2}
.
7. 2. Let Q(x’, \mu, \sigma) be the function defined in Lemma 6. 1 ) (i) and
assume the conditions (II) , . Then it follows from Lemmas 6. 4 (i),
6. 1 ) (i) and 6. 5 (or Remark 6. 1) that for (x, \mu, \sigma)\in U(x^{0})\cross U (O, )
\beta
\beta)
\gamma)
\sigma^{0}
\beta
is real valued for real
(7. 5)
Q(x’, \mu, \sigma)
(7. 6)
Q(x^{0}, O, \sigma^{0})=O
(7. 7)
,
,
-Q(x’, O, \sigma)\geq O
or
\mu
\leq 0
in the case (a),
in the case (b).
Let Q(x’, D’) be the operator whose symbol is the extension
. We consider the following boundary value problem
of
(P_{II}, Q) for
system:
Q(x’, \tau\neg,a)\in S_{+}^{0}
Q(x’, \tau-\theta(x, \sigma), \sigma)
2\cross 2
P_{II}(x, D)u=(D_{n}-M_{II}(x, D’)).t(u_{1}, u_{2})
(7.9)(7.8)
(P_{II}, Q)\{
=f(x) =t(f_{1}, f_{2})
Q(x’, D’)u_{1}(x’, O)+u_{2}(x’, O)=g(x’)
in
on
R_{+}^{n+1}
R^{n}
.
,
On structures of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
133
THEORE M7.1 . Assume the conditions (7. 5), (7. 6) and (7. 7). \cap Thm
there exist constants C,
such that for every
and u(x) the following a priori estimate for the problem (P_{II}, Q) holds:
\gamma_{0}>O
(P_{II}, Q)
\gamma\geq\gamma_{0}
||P_{II}u||_{0,\gamma}+ |Qu1+u_{2}|_{\frac{1}{2},\gamma}\geq C\gamma
||u||_{0,\gamma}
.
To prove this we need the following lemmas. The technique used in
the proof is also used in \S 8 for the proof of (1. 1).
Lemma 7. 2. Let (7. 8) be satisfified for u(x) . Then for every \delta>O
one can choose {or the size of supp ) and a constant
such that
u(x)
and
for every
\psi’
\epsilon_{0}
\mathcal{T}_{0}>0
\gamma\geq\gamma_{0}
(7. 10)
|\Lambda^{-\frac{1}{2}}u_{1}|_{0}^{2},v\leq\delta||u_{1}||_{0,\gamma}^{2}+\delta^{-1}||u_{2}||_{0,\gamma}^{2}+\delta^{-1}||\Lambda^{-1}f_{1}||_{0,\gamma}^{2}
consequmtly, let
\gamma_{0}(D’)
(7. 11)
be the operator with its symbol
\backslash ^{\gamma_{0}(D’)u_{1}}/
,
\tilde{\mathcal{T}}_{0}(\tau\Lambda_{\gamma}^{-1}, \sigma\Lambda_{\gamma}^{-1})\in S_{+}^{0}
, thm
\leq\gamma\delta||u1||_{0,\gamma}^{2}+\gamma\delta^{-1}||u_{2}||_{0,\gamma}^{2}
u_{1}/0 v
,
+\gamma\delta^{-1}||\Lambda^{-1}f_{1}||_{0,\gamma}^{2}
LEMMA 7. 3. Let R_{0}(x’, D’) be the one in Lemma 7. 1 and assume
(7. 9). Thm there exist constants C>0 ,
such that for any \delta>0 ,
and u(x)
\gamma_{0}>0
{\rm Im}/R_{0}(\backslash x’, D’)u
,
u/0 ,
\gamma\geq\gamma_{0}
\gamma
\geq-C((\delta\gamma)^{-1}|g|_{\frac{21}{2},r,\gamma}+\delta\gamma|u_{1}|_{-_{2}}^{2}\perp+|u_{1}|_{-\xi,r}^{2})
+ {\rm
where the symbol
of
R_{1}\in S_{+}^{0}
Re} R_{1}(x’, D’)u_{1}
,
,
u_{1/0}^{\backslash }
,
is represmted in
\gamma
U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}})
by
R_{1}(x’, \tau’, \sigma’)=-2d_{1}{\rm Re} Q+d_{2}|Q|^{2}+\epsilon’b+2T’f{\rm Im} Q
(7. 12)
=d_{1} (-2{\rm Re} Q+(1+\epsilon’e_{12})^{-1}(e_{11}-e_{22})\epsilon’)
+d_{2} (|Q|^{2}+(1+\epsilon’e_{12})^{-1}e_{21}\epsilon’)+2\mathcal{T}’f{\rm Im} Q
( Q is allowed to take complex values.)
LEMMA 7. 4. Assume the same conditions as in Theorem 7. 1 and let
(7. 9) be satisfified for u(x) . Then one can choose
(especially
and
C
,
,
constants
exist
that
there
such
so that
C’
and
for every
R_{1}\in S_{+}^{0}
d_{1}
\gamma_{0}>0
d_{2})
\gamma\geq\gamma_{0}
u(x)
{\rm Re}/R_{1}(\backslash x’, D’)u_{1}
Here C’ depends only on
,
u_{1}/0
,
\gamma\geq-C
(x^{0}, \tau^{0}, \sigma^{0})
|u1|_{1}^{2}-_{2},\gamma-C"/\gamma_{0}(\backslash D’)u_{1}
f
and ,
d_{1}
,
d_{2}
u_{1}/0
,
.
\gamma
but not on the size
support of .
Using these lemmas we first prove Theorem 7. 1.
\psi’
,
of the
The proofs of
T. Ohkubo and T. Shirota
134
these lemmas are given after that.
so that it satisfies (7. 1. 1)
PROOF OF THEOREM 7. 1. First choose
and (7. 4. 10) and the expression (7. 4. 7) are nonnegative. Secondly choose
satisfying (7. 4. 5) and (7. 4. 6) with C_{0}>1 . We
fix
and . Then Lemma 7. 3 and 7. 4 are valid for every with their
is chosen so that it satisfies
constants depending on . Furthermore if
(7. 1. 2), Lemma 7. 1 is valid and we obtain
d_{1}
d_{2}(x, \epsilon’, \sigma’)=d_{2}^{(0)}(x’, \sigma’)+\epsilon’d_{2}^{(1)}
f
d_{2}
d_{1}
f
f
||P_{II}u||_{0,\gamma}^{2}
\geq C(r^{2}(||u_{1}||_{0,\gamma}^{2}+C’||u_{2}||_{0,\gamma}^{2})+\gamma {\rm Im} \langle R_{0}u, u\rangle_{0,\gamma})
\geq C(\gamma^{2}(||u_{1}||_{0,\gamma}^{2}+C’||u2||_{0}^{2},\gamma)-C" r^{/}r_{0}(D’)u_{1}
,
u_{1}/t)
,
\gamma
-c_{1}(\delta_{1}^{-1}|g|_{1,}^{2}+\delta_{1}\mathcal{T}^{2}|u_{1}|_{-5^{\gamma}z\prime}^{2},+\mathcal{T}|u_{1}|_{-l_{\gamma}}^{2})2^{f}
-C_{2}\gamma|u_{1}|_{-\perp}^{2}\gamma)2
”
f
can be taken arbitrarily.
whepe , , C’ and C’ depend on and
Hence it follows from (7. 10) with fixed that
C_{2}
C_{1}
\delta_{1}>0
\delta
C(||P_{II}u||_{0,\gamma}^{2}+C_{1}\delta_{1}^{-1}|g|_{1}^{2},\gamma)2
\geq T^{2}(1-\delta_{1}C_{3}-C_{4}\gamma^{-1})||u1||_{0}^{2}
,
\gamma
+r^{2}(C’-\delta_{1}C_{3}-C_{4}\gamma^{-1})||u2||_{0}^{2}
-C’\gamma_{\backslash }\gamma_{0}(/D’)u_{1}
and
depending on
where
C_{4}
C_{3}
depend on
f.
,
\gamma
,
u_{1/0}^{\backslash }
,
,
\gamma
Using (7. 11) we obtain for any
\delta>0
S_{0}
C(||P_{II}u||_{0,\gamma}^{2}+|g|_{1}^{2},\gamma)2
\geq \mathcal{T}^{2}(1-\delta_{1}C_{3}-C_{4}\mathcal{T}^{-1}-\delta C’)||u1||_{0,r}^{2}
+\gamma 2(C’-\delta_{1}C_{3}-C_{4}\gamma^{-1}-\delta^{-1}C")
\equiv\gamma^{2}(C_{5}||u_{1}||_{0,\gamma}^{2}+C_{6}||u_{2}||_{0,\gamma}^{2})
||u_{2}||_{0}^{2}
,
\gamma
.
are fixed we see from (7. 1. 3) and (7. 4. 11) that in order
and
positive it suffices to choose , and
so that
and
to make
Since
d_{2}
d_{1}
f
C_{6}
C_{5}
\delta
\delta_{1}
1-\delta_{1}C_{3}-C_{4}\mathcal{T}^{-1}-\delta|2f{\rm Im} Q-2d_{2}{\rm Im} Q^{(1)}|>0,\cdot
and
f-\delta_{1}C_{3}-C_{4}\gamma^{-1}-\delta^{-1}|2f{\rm Im} Q-2d_{2}{\rm Im} Q^{(1)}|
is sufficiently large.
The process of the choice is as follows:
(1)
Choose
\delta
(hence ) such that
\epsilon_{0}
\delta
.
|2d_{2}
{\rm Im} Q^{(1)}|< \frac{1}{4}
and fix the
\delta
.
On structures of certain
L^{2}
f
-well-posed mixed problems for hyperbolic systems
Choose so that
is sufficiently large.
(3) Since Q(x^{0}, O, \sigma^{0})=O we can choose such a small
(2)
135
f-\delta^{-1}|2d_{2}{\rm Im} Q^{(1)}|
\epsilon_{0}
that in
S_{8\text{\’{e}}_{0}}(\tau^{0}, \sigma^{0})
\delta
.
|2f{\rm Im} Q|< \frac{1}{4}
,
remains large.
f-\delta^{-1}|2f{\rm Im} Q|-\delta^{-1}|2d_{2}{\rm Im} Q^{(1)}|
Choose
(4)
and
\delta_{1}
\gamma_{0}
so that
\delta_{1}
C_{3}+C_{4} \gamma_{0}^{-1}<\frac{1}{4}
.
larger than -and complete the proof.
and
Thus we obtain
PROOF OF LEMMA 7. 2. From integration by parts it holds that
C_{5}
C_{6}
\frac{1}{4}
|\Lambda^{-\frac{1}{2}}u_{1}|_{0,\gamma}^{2}=\langle\Lambda^{-1}u_{1}, u_{1}\rangle_{0,\sim}=2{\rm Im}(\Lambda^{-1}D_{n}u_{1}, u_{1})_{0,\gamma}(
It follows from Lemma 3. 2 and (7. 8) that
D_{n}u_{1}=\lambda_{1}^{(0)}u_{1}+\Lambda_{0}^{(0)}u_{2}+(\epsilon’-i\gamma’)p_{11}\Lambda u_{1}+(\epsilon’-i\mathcal{T}’)p_{12}\Lambda u_{2}+f_{1}
are of order O and the symbol of
where ), , r’,p_{11} and
and real. Hence we see from Lemma 2. 1 ) (iv) that
\lambda_{1}^{(0}
\epsilon’
p_{12}
\lambda_{1}^{(0}
)
,
is
\lambda_{1}(x, O,\tilde{\sigma})
\alpha
|\Lambda^{z}-1u_{1}|_{0,\gamma}^{2}\leq 2{\rm Im}
\{(\Lambda_{0}^{(0)}\Lambda^{-1} u2’ u_{1})_{0,\sqrt}+ (\Lambda^{-1}
+((\epsilon’-i\gamma’)p_{11}u_{1}
+
C\gamma^{-1}||u||_{0}^{2}
,
\gamma
,
fl,
u_{1})_{0,\gamma}
u_{1})_{0,*}+((\epsilon’-i\gamma’)p_{12}u_{2},
u_{1})_{0,\gamma}\}
,
where the last term of the right arises from commutators and C>0 depend
we see from ( 2. 6)’ that
only on
. Since
M_{II}
(\tilde{x},\tilde{\tau},\tilde{\sigma})\in S_{\epsilon,\tau^{*}0}(x^{0})\cross S_{8\text{\’{e}}_{0}}(\tau^{0}, \sigma^{0})
|\epsilon
’
(\tilde{x},\tilde{\tau},\tilde{\sigma})|\leq 8\epsilon_{0}
and
7’(\tilde{\tau},\tilde{\sigma})\leq
4e0 .
is
Consequently the first inequality follows from Lemma 2. 1 ) (vi) if
taken sufficiently small. The second one follows from the first one because
.
of
Proof 0F Lemma 7. 3. From (7. 9) we have
\beta
\epsilon_{0}
\gamma_{0}(D’)\cdot\psi(D’)=\gamma_{0}(D’)\cdot\Lambda\cdot\Lambda^{-1}\cdot\psi(D’)=r\Lambda^{-1}\psi(D’)
u_{2}(x’, O)=g(x’)-Q(x’, D’)u_{1}(x’, O)(
Hence it follows from the definition of
{\rm Im} R_{0}(x’, D’)u
=2d_{1}
,
u/0 ,
R_{0}
that
\gamma
{\rm Re}\langle g-Qu_{1}, u_{1}\rangle_{0,\gamma}+{\rm Re}/d_{2}\backslash \cdot(g-Qu_{1})
-2\gamma f{\rm Im}\Lambda^{-1}\cdot(g-Qu_{1})
,
,
g-Qu_{1/0,\gamma}\backslash
u_{1/0,\gamma}+R\backslash e_{\backslash }(/\epsilon b)u_{1}
,
u_{1}/0
,
\gamma
T. Ohkubo and T. Shirota
136
=2d_{1}
{\rm Re}
\langle g, u_{1}\rangle_{0,\gamma}+{\rm Re}
-{\rm Re}
\langle d_{2}g, g\rangle_{0,\gamma}-2\gamma f{\rm Im}\langle\Lambda^{-1} g, u_{1}\rangle_{0,\gamma}
\langle d_{2}\cdot Qu_{1}, g\rangle_{0,\gamma}-{\rm Re}
-2d_{1}{\rm Re}
+2\gamma f
\langle d_{2}g, Qu_{1}\rangle_{0,\gamma}
\langle d_{2}\cdot Qu_{1}, Qu_{1}\rangle_{0,\gamma}
\langle Qu_{1}, u_{1}\rangle_{0,\gamma}+{\rm Re}
{\rm Im}
,
\langle 4^{-1}\cdot Qu_{1}, u_{1}\rangle_{0,\gamma}+{\rm Re}/(\backslash \epsilon b)u_{1}
u_{1/0}^{\backslash }
,
\gamma
, we have the following estimates with respect to the
Noting that ,
first five terms of the right hand side of the above equality:
d_{2}
Q\in S_{+}^{0}
|\langle g, u_{1}\rangle_{0,\gamma}|=|\langle\Lambda^{\frac{1}{2}}g, \Lambda^{-\frac{1}{2}}u_{1}\rangle_{0,\gamma}|
\leq
|g| \frac{1}{2}
,
\gamma|u1|_{-\frac{1}{2}}
,
\leq\delta r
|u1|_{-1}^{2}\gamma+2
\gamma
(\delta\gamma)^{- 1}|g|_{1}^{2}2
’
|\langle d_{2}g, g\rangle_{0,\gamma}|\leq C|g|_{0,\gamma}^{2}\leq C\gamma^{-1}|g|_{z}^{2}\perp,\gamma
,
\gamma
,
,
|\gamma\langle\Lambda^{-1} g, u_{1}\rangle_{0,\gamma}1\leq\gamma|g|_{-\frac{1}{2}\gamma},|u_{1}|_{-1}2’\gamma
\leq\delta\gamma
|u1|_{-_{2}}^{2}1
, \gamma+
.
|\langle d_{2}\cdot Qu_{1},g\rangle_{0,\gamma}|’
\leq\delta r
(\delta\gamma)^{-1}
\geq- C(d_{1},
\gamma
,
( r)-1lgl
\delta
|u1|_{-1}^{2}\gamma+2
Hence we obtain for large
,
,
|\langle d_{2}g, Qu_{1}\rangle_{0,\gamma}|\leq C|g|_{1}\gamma|u_{1}|_{-1}2’ 2’\gamma
’
{\rm Im} R_{0}(x’, D’)u
|g| \frac{21}{2}
21_{\gamma}\tau 2
’
\gamma
u_{/0}^{\backslash }
,
\gamma
d_{2}, f)
\cdot(\delta\gamma|u_{1}|_{1}^{2}-_{2},r+(\delta\gamma)^{-1}|g|_{z}^{2}\perp,)r
-2d_{1}{\rm Re} \langle Qu_{1}, u_{1}\rangle_{0,r}+{\rm Re}(d_{2}Q^{*}Q)u_{1}
+2\gamma f
{\rm Im}^{/}(Q\Lambda^{-1})u_{1}
-C(d_{2}, f, Q)|u1|_{-\frac{1}{2}}^{2}
,
u_{1/_{0,r}}.+R\backslash e_{\backslash }(/\epsilon b)u_{1}
,
,
u_{1/0,\gamma}^{\backslash }
u_{1}/0
,
\gamma
,r
where the last term described above arises from the commutators. Thus
the lemma follows from (7. 4).
ProoF 0F Lemma 7. 4. For the sake of the generalization (see subsection 10. 2), assume that Q is complex valued. From Taylor’s expansion
we have
Q(x’, \mu’, \sigma’)=Q^{(0)}(x’, \sigma’)+\mu’Q^{(1)}(x’, \sigma’)
+ (\mu’)^{2}Q^{(2)}(x’, \mu’, \sigma’)
e_{ij}
(
where
(x’, \epsilon’, \sigma’)=e_{ij}^{(0)}(x’, \sigma’)+\epsilon’e_{ij}^{(1)}(x’, \epsilon’, \sigma’)
1+\epsilon’e_{12}(x’, \epsilon’, \sigma’)
Q^{(0)}=Q(x’, O, \sigma’)
,
)- 1=
,
1-\epsilon ’ e_{12}^{(0)}(x’, \sigma’) +O((\epsilon’)^{2})
Q^{(1)}= \frac{\partial Q}{\partial\mu}(x’, O, \sigma’)
and
(i,j=1,2) ,
,
,
e_{ij}^{(0)}=e_{if}(x’,\cdot O, \sigma’)
. Since
\mu’
On structures
=
\epsilon’-i\gamma’
of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
, from the definition (7. 12) of
R_{1}
we see that for
137
(x’, \tau’, \sigma’)\in U(x^{0})
\cross(U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}})
R_{1}(x’, \mu’, \sigma’)=
(r’
f
{\rm Im} Q
-2d_{1}({\rm Re} Q^{(0)}+\epsilon’{\rm Re} Q^{(1)}+\gamma’{\rm Im} Q^{(1)}
+((\epsilon’)^{2}-(\gamma’)^{2}) {\rm Re} Q^{(2)}+2\epsilon’\gamma’{\rm Im} Q^{(2)})
+d_{2}\{|Q^{(0)}|^{2}+((\epsilon’)^{2}+(r’)^{2})|Q^{(1)}|^{2}+((\epsilon’)^{2}+(r’)^{2})^{2}|Q^{(2)}|^{2}
+\epsilon
’
+2
{\rm Re} (\epsilon’+i\gamma’)
+2
{\rm Re}((\epsilon’)^{2}-(\gamma’)^{2}+i2\epsilon’\gamma’)\overline{Q^{(2)}}Q^{(0)}
+2
{\rm Re}(\epsilon’-i\gamma’)((\epsilon’)^{2}-(\gamma’)^{2}+i2\epsilon’r’)\overline{Q^{(2)}}Q^{(1)}\}
\overline{Q^{(1)}}Q^{(0)}
(1-\epsilon’e_{12}^{(0)}+o((\epsilon’)^{2}))\{d_{1}(e_{11}^{(0)}-e_{22}^{(0)})+d_{2}e_{21}^{(0)}
+d_{1}\epsilon’(e_{11}^{(1)}-e_{22}^{(1)}) +d_{2}\epsilon’ e_{21}^{(1)}\}
Put
Then the above equals to
d_{2}(x, \epsilon’-i\gamma’, \sigma’)=d_{2}^{(0)}(x’, \sigma’)+\epsilon’d_{2}^{(1)}
=r’
.
with some real
d_{2}^{(0)}
and
C_{1}+(-2d_{1}{\rm Re} Q^{(0)}+d_{2}^{(0)}|Q^{(0)}|^{2})
+\epsilon’(d_{2}^{(1)}|Q^{(0)}|^{2}-d_{1}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)}))
(7. 4. 1)
+d_{2}^{(0)}(e_{21}^{(0)}+2{\rm Re}\overline{Q^{(1)}}Q^{(0)}))
+
(\epsilon’)^{2}(d_{2}^{(1)}(e_{21}^{(0)}+2{\rm Re}\overline{Q^{(1)}}Q^{(0)})+k_{1}d_{1}+k_{2}d_{2}^{(0)})
+O
(
|\epsilon’|^{3}
),
where
C_{1}=2f{\rm Im} Q-2d_{1}({\rm Im} Q^{(1)}+2\epsilon’{\rm Im} Q^{(2)}-\gamma’{\rm Re} Q^{(2)})
+d_{2}\{-2
+\gamma
’
+2
and
k_{1}
,
k_{2}
(2
{\rm Im}\overline{Q^{(1)}}Q^{(0)}+2{\rm Re}
(\epsilon’)^{2}+(\mathcal{T}’)^{2}
{\rm Re}
)
(-\gamma’+i2\epsilon’)
|Q^{(2)}|^{2}+\mathcal{T}’|Q^{(1)}|^{2}
(-i\overline{(\mu’)}^{2}-\epsilon’\mathcal{T}’+i2(\epsilon’)^{2}
do not depend on
d_{1}
,
\overline{Q^{(2)}}Q^{(0)}
d_{2}
and
)
\overline{Q^{(2)}}Q^{(1)}\}
,
f:
k_{1}=-2{\rm Re} Q^{(2)}-e_{12}^{(0)}(e_{11}^{(0)}-e_{22}^{(0)})+e_{11}^{(1)}-e_{22}^{(1)}
k_{2}=|Q^{(1)}|^{2}+2{\rm Re}\overline{Q^{(2)}}Q^{(0\dot{)}}-e_{12}^{(0)}e_{21}^{(0)}+
Now we shall determine
(7. 4. 2)
d_{1}
,
d_{2}^{(0)}
and
d_{2}^{(1)}
e_{21}^{(1)}
as follows:
-2d_{1}{\rm Re} Q^{(0)}+d_{2}^{(0)}|Q^{(0)}|^{2}\geq 0
,
.
,
d_{2}^{(1)}
const).
T. Ohkubo and T. Shirota
138
(7’. 4.3)
the coefficient of
(7. 4. 4)
the coefficient of
\epsilon’
is zero,
(\epsilon’)^{2}\equiv C_{0}>1
From (7. 4. 3) we obtain
(7. 4. 5)
d_{2}^{(0)}=-k_{3}^{-1}(d_{2}^{(1)}|Q^{(0)}|^{2}-d_{1}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)})))
where
does not vanish because of
Insert this into , then
k_{3}\equiv e_{21}^{(0)}+2{\rm Re}\overline{Q^{(1)}}Q^{(0)}
Q^{(0)}(x^{0}, \sigma^{0})=0.
e_{21}^{(0}
,
) \neq 0
and
((7.2))
C_{0}
(7. 4. 6)
C_{0}=d_{2}^{(1)}(k_{3}-k_{2}k_{3}^{-1}|Q^{(0)}|^{2})
+d_{1}(k_{1}+k_{2}k_{3}^{-1}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)})))
.
so that (7. 4. 2) is fulfilled.
Let us show that it is possible to choose
From (7. 4. 5) and (7. 4. 6) we have for some fixed C_{0}>1
d_{1}
-2d_{1}{\rm Re} Q^{(0)}+d_{2}^{(0)}|Q^{(0)}|^{2}
(7. 4. 7)
=d_{1}(-2{\rm Re} Q^{(0)}+k_{3}^{-1}|Q^{(0)}|^{2}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)})))
-k_{3}^{-1}|Q^{(0)}|^{4}d_{2}^{(1)}
=k_{3}^{-2} (1-k_{3}^{-2}k_{2}|Q^{(0)}|^{2})^{-1}(k_{4}d_{1}-C_{0}|Q^{(0)}|^{4})
,
where
k_{4}=-2k_{3}^{2}{\rm Re} Q^{(0)}+|Q^{(0)}|^{2}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)}))k_{3}
+|Q^{(0)}|^{2}(2k_{2}{\rm Re} Q^{(0)}+k_{1}|Q^{(0)}|^{2})
The condition (7. 5) implies that for any small
such that
borhood
\delta>O
.
there exists a neigh-
U(x^{0})\cross U(\sigma^{0})
(7. 4. 8)
\delta|{\rm Re}
Q^{(0)}|\geq|Q^{(0)}|^{2}\geq|Q^{(0)}|^{4}
in
U(x^{0})\cross U(\sigma^{0})
.
in
U(x^{0})\cross U(\sigma^{0})
.
Hence it holds for some C>0 that
(7. 4. 9)
Note that for
(7. 4. 10)
C_{0}|Q^{(0)}|^{4}\leq C|k_{4}|
d_{1}
satifying (7. 1. 1) we have from (7. 2) and (7. 7)
k_{4}d_{1}\geq-Cd_{1}{\rm Re} Q^{(0)}\geq 0
with some C>O , because of ) =
and (7. 4. 10) there exists a real number
e_{12}^{(0}
h_{21}(x, O, \sigma)
k_{4}d_{1}-C_{0}|Q^{(0)}|^{4}\geq 0
Since
(7. 4. 2.)
k_{3}^{-2}(1-k_{3}^{-2}k_{2}|Q^{(0)}|^{2})^{-1}
d_{1}
. Then it is seen from
(7. 4. 9)
(satisfying (7. 1. 1)) such that
in
U(x^{0})\cross U(\sigma^{0})
in (7. 4. 7) is positive, we thus obtain
d_{1}
.
satisfying
On structures of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
139
Since
and Q(x^{0}, O, \sigma^{0})=O , a real number.d_{2}^{(1)} i s determined by
(7. 4. 6) so that (7. 4. 4) is valid. Take
satisfying (7. 4. 5), then (7. 4. 2),
(7. 4. 3) and (7. 4. 4) hold. Furthermore
does not
depend on . Thus we see from (7. 4. 1) that
k_{3}\neq 0
d_{2}^{(0)}
-2d_{1}{\rm Re} Q^{(0)}+d_{2}^{(0)}|Q^{(0)}|^{2}
\epsilon’
R_{1}(x’, \mu’, \sigma’)-\gamma\prime c_{1}\geq O
in a sufficiently small neighborhood
Finally choose
so that
\sigma)\in R
(O,
S_{8\cdot 0}(\tau^{0}, \sigma^{0})\subset U(\tau^{0}, \sigma^{0})
\epsilon_{0}
R_{1}(x’D’) ,
U(x^{0})\cross U
R_{1}(\tilde{x}(x’),\tilde{\tau}(\tau\Lambda_{f}^{-1}, \sigma\Lambda_{\gamma}^{1}),\tilde{\sigma}(\tau\Lambda_{\gamma}^{1}, \sigma\Lambda_{\gamma}^{1}))
\sigma^{0}
).
. Then for the symbol of
, it holds that for every (x’, ,
\tau
\cross C_{-}\cross R^{n-1}
R_{1}(\tilde{x},\tilde{\tau},\tilde{\sigma})+C"\tilde{\mathcal{T}}\geq 0
,
where C’>0 is a constant such that for all
(7.4.11)
C’\geq|C_{1}|
(x’, \tau^{j}, \sigma’)\in S_{5}.(x^{0})\cross S_{8}.(\tau^{0}, \sigma^{0})\pi 00
=|C_{1}(x’, \tau,\sigma’’ ; f, d_{1}, d_{2})|
.
Hence the assertion of Lemma 7. 4 follows from Lemma 2. 1 ).
REMARK 7. 1. It is seen from the above proof and (7. 12) that the
assertion of Theorem 7. 1 is also valid if it satisfies (7. 1. 1) and
\beta
d_{1}(-2{\rm Re} Q+(1+\epsilon’e_{12})^{-1}(e_{11}-e_{22})\epsilon’)+d_{2}(|Q|^{2}+(1+\epsilon’e_{12})^{-1}e_{21}\epsilon’)\geq O
for any
(x’, \epsilon’, \mathcal{T}’)
in a real neighborhood U of
(x^{0}, O, \sigma^{0})
such that
(\epsilon’+\theta\{xf
,
\gamma’))^{2}+|\sigma’|^{2}\leq 1.
over U, we see
Furthermore using
with some
that the above condition is equivalent to that on U
if |Q|^{2}+
in the case (a), which follow
on U.
from that
k\in S_{+}^{0}
(\begin{array}{ll}1 kO 1\end{array})M_{II}(\begin{array}{ll}1 kO 1\end{array})
{\rm Re} Q\leq 0
(1 + \epsilon’ e_{12})^{-1}e_{21}
\epsilon’=O
{\rm Re} D(x’, \sigma’)\geq O
\S 8. The proof of Theorem 1. 1
In \S 6 and the latter part of \S 5 we have tried
to analyse how the
-well-posedness of the freezing problem dominates the relations among
the coefficients of the boundary operator. In this section we prove (1. 1)
on the base of that analysis in three subsections. In order to prove (1. 1)
for k=0 we have only to show (3. 4) whether the set II is empty or not.
In subsection 8. 1 we prove (3. 4) by the consideration in \S 5, for the case
where the set II is empty. In subsection 8. 2 we do so by that in \S 6
together with the methods used in \S 7, for the case where the set II is
in (3. 4) on support of does
not empty. Note that the dependence of
not affect the validity of Lemma 3. 3. The proof of (1. 1) for k\geq 1 is given
in subsection 8. 3.
8. 1. We prove (3. 4) in the case where the set II is empty, using the
L^{2}
C_{1}
\psi
T. Ohkubo and T. Shirota
140
estimates
From
(P_{I}^{\pm})
,
(P_{I}^{\pm})
and
(P_{III}^{\pm})
and
in \S 5.
it holds for
(B^{+})
(P_{III}^{\pm})
that
U\in H_{1,\gamma}(R_{+}^{n+1})
C\gamma||U||_{0,\gamma}
\leq C\gamma(||u_{I}^{+}||_{0,\gamma}+||u_{I}^{-}||_{0,\gamma}+
lluI+II
||0
,
r+||u_{III}^{-}||_{0,\gamma})
(8. 1)
\leq||P_{I} u_{I}||0
+\gamma^{\frac{1}{2}}|u_{I}
,
\gamma+||P_{III}^{+}u_{III}^{+}||_{0,r}+||P_{I}^{-}u_{I}^{-}||_{0,\gamma}+||P_{III}^{-}u_{III}^{-}||_{0,f}
|0 , r+\gamma|u_{II}^{+} |_{-1},r
I
On the other hand it follows from
.
(B_{1})
and Lemma 5. 3 (ii) that
g=V_{I}^{+}(x’, D’)u_{I}^{+}(x’, O)+V_{III}^{+}(x’, D’)u_{III}^{+}(x’, O)
+B^{+}
(x’, D’)
C_{I}(x’, D’)u_{I}^{-}(x’, O)
\cdot
+K_{I}(x’, D’)u_{I}^{-}(x’, O)+V_{III}^{-}(x’, D’)u_{III}^{-}(x’, O)
,
that is,
(V_{I}^{+}, V_{III}^{+})[
=
(\begin{array}{l}u_{I}^{+}(O)u_{III}^{+}(O)\end{array})+C_{I}u_{I}^{-}(O)]
,
g-V_{III}^{-}u_{III}^{-}(0)-K_{I}u_{I}^{-}(O)
where
K_{I}(x’, D’)
=(B^{+}C_{1})
(x’, D’)-B^{+}(x’, D’)\cdot
C_{I}(x’, D’)\in S_{+}^{-1}
matrix
: (m-l)\cross l matrix.
:
C_{I}^{1}
C_{I}(x’, D’)=(\begin{array}{l}C_{I}^{1}C_{I}^{3}\end{array})
with
\in S_{+}^{0}
\{
C_{I}^{3}
Using the estimates
(B^{+})
and
(P_{III}^{-})
,
l\cross l
we have
C(\gamma^{1}2|u_{I} +C_{I}u_{I}^{-}|_{0,\gamma}+\gamma|u_{III}^{+} +C_{I}u_{I}^{-}|_{-\perp}.2,f)
\leq|g|_{\frac{1}{2}}
\leq|g|1
Hence for large
\gamma
,
\gamma+|V_{III}^{-}u_{III}^{-}|_{1}+|K_{I}u_{I}^{-}|_{1}z
,
r^{+||P_{III}^{-}u_{III}^{-}||_{0,\gamma}+|u_{I}^{-}|_{-_{2}^{1},r}}
”
2 ,r
.
it follows from the estimate
C(\gamma 3_{1}u_{I}
|0
, r^{+\gamma|}
\leq|g|1
\leq|g|1
,
,
uI+II
|_{-}1
,
(P_{I}^{-})
that
f)
r+||P_{III}^{-}u_{III}^{-}||_{0,\gamma}+\gamma^{\frac{1}{2}}|uI-|0
,r
\tau^{+||P_{I}^{-}u_{I}^{-}||_{0,r}+||P_{III}u_{III}}--||_{0,\gamma}
.
Combining this with (8. 1) we obtain (3. 4).
8. 2. In this subsection we prove (3. 4) in the case where the set II
the proof is the same as that in [7],
is not empty. If
,
,
and the
that is, (3. 4) is obtained by using the estimates
R(x^{0}, \tau^{0}, \sigma^{0})\neq 0,
(P_{I}^{\pm})
(P_{III}^{\pm})
(P_{II})’
On structures
relation
of certai n
L^{2}
-well-posed mixed problems for hyperbolic systems
141
together with Lemma 2. 1 ) (ii) and taking the constant
in
suitably. Therefore we prove it in the case where
by using the estimates
,
,
and Lemmas in \S 6 (especially Lemma 6. 2 and 6. 7).
From Lemma 6. 2 it holds that for small
(B_{1})
C_{1}
\beta
(P_{II})’
R(x^{0}, \tau^{0}, \sigma^{0})=O,
(P_{I}^{\pm})
(P_{II})
(P_{III}^{\pm})
\delta_{1}>0
|g-V_{III}^{-}u_{III}^{-}
-Cl
u_{I}^{-}-C_{2}u_{\acute{I}}I|_{\frac{z_{1}}{2}.\gamma}
\geq C\{\gamma(|g_{1}|_{0,\gamma}^{2}+|g_{2}|_{0,f}^{2})+\delta_{1}r^{2}
where
C_{1}
,
|g_{3}|_{-1}^{2}2
\gamma\}
,
,
and
C_{2}\in S_{+}^{-1}
|0
|g_{1}|_{0,r}^{2}\geq|u_{I}
,
\gamma-|k_{I}
1
u_{I}^{-}|_{0,\gamma}^{2}-|k_{I}
|g_{2}|_{0,r}^{2}=|u_{II}’+QuII +k_{II}
1u_{I}^{-}|_{0,\gamma}^{2}
II
,
u_{II}’|_{0,r}^{2}
,
|g_{3}|_{-_{2}}^{2}z,\geq|ru_{III}^{+}|_{-\frac{1}{2}}^{2},r-|k_{III1}u_{I}^{-}|_{-1}^{2}rz’-|k_{III}
Since the symbols of
II
u_{II}’|_{-\not\equiv,\gamma}^{2}
.
,
k_{JK}\in S_{+}^{0}
C(|g|_{\frac{21}{2},r}+|u_{III}^{-}|_{\frac{21}{2},r})
\geq\gamma
|u_{I}
|_{0}^{2}
, r^{+\delta_{1}\gamma|}2
-C(\gamma|uI-|_{0}^{z}
,
uI+II
(P_{I}^{\pm})
,
(P_{III}^{\pm})
’
r
\gamma+\delta 1\mathcal{T}^{2}|uI-|_{-\frac{1}{2}}^{2}
-\gamma|k_{I}IIu_{II}’|_{0,f}^{2}
Using the estimates
|^{2}-z
,
r+\delta_{1}\gamma^{2}|u_{II}|_{-\not\in}^{2}
,
\gamma
)
.
we have for large
\gamma
C(||P_{I}^{+}u_{I}^{+}||_{0,\gamma}^{2}+||P_{I}^{-}u_{I}^{-}||_{0,f}^{2}+||P_{III}^{+}u_{III}^{+}||_{0,\gamma}^{2}+||P_{III}^{-}u_{III}^{-}||_{0,r}^{2}+|g|_{\S,r}^{2})
\geq\gamma 2(||u_{I}^{+}||_{0,r}^{2}+||uI-||_{0,\gamma}^{2}+\delta 1
-\gamma|k_{I}IIu’II|_{0}^{2}
,
||u_{III}^{+}||_{0,\gamma}^{2}+||u_{III}^{-}||_{0}^{2}
\gamma-C\delta 1\gamma^{2} |u_{II}|_{-1}^{2}2’\gamma
, r)
.
From Lemma 7. 2 we have
(8. 2)
,\gamma\leq C^{2}(\gamma^{-2}||f_{II}’||_{0,\gamma}^{2}+||u_{II}||_{0,r}^{2})
|u_{II}|_{-}^{2}
-
Since
\delta_{1}
is small it follows from the estimate
||P_{1}
(8. 3)
.
(P_{II})
that
U||_{0,\gamma}^{2}+|g|_{\not\equiv,\gamma}^{2}
\geq cr^{2}||
U||_{0}^{2}
+C\gamma
,f
\{{\rm Im}
(Ro uu ,
u_{II}\rangle_{0,r}-|k_{I}
II
u_{II}’|_{0,r}^{2}\}
On the other hand from the boundary condition
(V_{I}^{+}, V_{II}’, V_{II}^{+})(x’, D’)\cdot
t(tuI+, u_{II}^{\prime\prime t},uIII+)
=g (x’)-(V_{I}^{-}, V_{II}’, V_{III}^{-})(x’, D’)\cdot
Since
\det(V_{I}^{+}, V_{II}’, V_{II}^{+})(x^{0}, \tau^{0}, \sigma^{0})\neq 0
(B_{1})
.
we have
(x’)
t(tu_{I}^{-}, u_{II}^{\prime t},u_{III}^{-})
(x’)
.
it follows from Lemma 2. 1
\beta
) (iii)
that
142
T. Ohkubo and T. Shirota
t(tuI+, u_{II}^{\prime\prime t},uIII+)
=C’ g-((V_{I}^{+}, V_{II}’, V_{III}^{+})^{-1}(x’, D’)+T’(x’, D’))
.
where
C’\in S_{+}^{0}
(8. 4)
and
(V_{I}^{-}, V_{II}’, V_{III}^{-})(x’, D’)\cdot
T’\in S_{+}^{-1}
t(tu_{I}^{-}, u_{II}’ , tu_{III}^{-})
. Hence from Lemma 6. 2 we have
u_{II}’(x’)=C_{1}g-
Tu’-kIII
u_{I}^{-}-k_{II}
II
u_{II}’-k_{II}
u_{III}^{-}
III
,
where
T=
(T_{I}, T_{II}, T_{III})\in S_{+}^{-1}
u’=t(tu_{I}^{-}, u_{II}^{\prime t},u_{III}^{-})
,
,
.
Q(x’, \tau, \sigma)
k_{IIII}^{\sim}(x’, \tau, \sigma)=
As in the proof of Lemma 7. 3, let us delete
we have
{\rm Im}
,
C_{1}\in S_{+}^{0}
by using (8. 4).
u_{II}’
Then
\langle R_{0}u_{II}, u_{II}\rangle_{0,\gamma}
=2d_{1}
\langle C_{1}g-Tu’-k_{II1}
{\rm Re}
+
{\rm Re}
\langle
d_{2}
.(
uI–QuI’
I^{-k_{II}}
,
III u_{III}^{-}
C_{1}g-Tu-\prime k_{II1}u_{I}^{-}-Qu_{II}’-k_{II}
, ( C_{1}g- Tu’-kIII u_{I}^{-}-Qu_{II}’-k_{II} III
III
u_{III}^{-}
)
u_{II}’\rangle_{0,r}
u_{III}^{-}
)
\rangle_{0,\gamma}
-2\gamma f{\rm Im} \langle 4^{-1}.(C_{1}g-Tu’-k_{II1}u_{I}^{-}-Qu_{II}’-k_{IIIII}u_{III}^{-}), u_{II}’\rangle_{0.\gamma}
+
{\rm Re}
\langle(\epsilon b)u_{II}’, u_{II}’\rangle_{0,\gamma}
.
, the estimates of the right hand terms of the
Since ,
and
above are as follows: for some constant C>O
C_{1}
T\in S_{+}^{-1}
k_{JX}\in S_{+}^{0}
|2d_{1}
\langle C_{1}g, u_{II}’\rangle_{0,\gamma}|\leq C(\delta\gamma|u_{II}’|_{-\frac{1}{2},r^{+(\delta r)^{-1}|g|_{z}^{2}},\gamma}^{2}\perp)
|2d_{1}
\langle
Tu’,
u_{II}’\rangle_{0,\gamma}|\leq C|u’|_{-_{\vec{2}}^{1},\gamma}\cdot|uII’|_{-\frac{1}{2}}
,
,
\gamma
\leq C(|u_{I}^{-}|_{-\frac{1}{2},f}+|u_{II}’|_{-\frac{1}{2}.\gamma}+|u_{III}^{-}|_{-\not\in,\gamma})|u_{II}’|_{-z’ r}1
\leq C
|2d_{1}
\langle k_{II}
III
u_{III}^{-}
,
(
u_{II}’\rangle_{0,\gamma}|\leq C.
\leq C
2d_{1}{\rm Re}
\gamma+|uII’|_{\frac{21}{2}}
|uI-|_{-\lrcorner}^{2}2
’
|u_{III}^{-}|_{\frac{1}{2},\gamma}\cdot
(
,
r^{+1}uIII-|_{-z}^{2}1
|u_{II}|_{-\frac{1}{2}}
\delta\gamma|u_{II}|_{-\not\in}^{2}
, \gamma+
,
{\rm Re}
\langle
(\delta\gamma)^{-1}
\leq 2d_{1}
{\rm Re}
\langle
\leq 1
.
|k11* 1
uI-,
k_{II1}^{\#}u_{II}’\rangle_{0,\gamma}
uI-,
k_{III}^{*}u_{II}’\rangle_{0,r}+C|u_{I}^{-}|_{0,\gamma}\cdot
|u_{III}^{-}|
|uII’|_{-1}
u_{II}’|_{0,\gamma}^{2}+C(|u_{I}^{-}|_{0,\gamma}^{2}+|u_{II}’|_{-1,\gamma}^{2})
),
\gamma
\langle k_{II1}u_{I}^{-}, u_{II}’\rangle_{0,r}
=2d_{1}
,r
,
,f
\S , f
),
On structures of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
|\langle d_{2}\cdot k_{III}u_{I}^{-}, k_{III1I}u_{III}^{-}\rangle_{0,\gamma}|\leq C(|u_{I}^{-}|_{0,\gamma}^{2}+|u_{III}^{-1_{0,\gamma)}^{2}}
|\langle d_{2}\cdot k_{II1}u_{I}^{-}, Qu_{II}’\rangle_{0,\gamma}|\leq|d_{2}
.
143
,
k_{II1}u_{I}^{-}|_{0,r}^{2}+1\cdot|Qu_{II}’|_{0,\gamma}^{2}
\leq C |uI-|0 f+1\cdot|Qu_{II}’|_{0.r}^{2}
,
,
|2\gamma f\langle\Lambda^{-1}\cdot C_{1}g, u_{II}’\rangle_{0,\gamma}|\leq Cr((\delta\gamma)^{-1}|g|_{0,\gamma}^{2}+\delta \mathcal{T}|u_{II}’|_{-1,\gamma}^{2})
,
\leq C(\delta^{-1}|g|_{0,\gamma}^{2}+\delta\gamma|u_{II}’|_{-\xi,r}^{2})
|2\gamma f\langle 4^{-1}.
’
Iu’, u_{II}’\rangle_{0,\gamma}|
-
1|
\leq C\gamma\{(\delta\gamma)
Tu’
|_{0}^{2}
, r+\delta\gamma
|uII’|_{-1}^{2}
,
\gamma
}
\leq C\{\delta^{-1}(|u_{I}^{-}|_{-1,\gamma}^{2}+|u_{II}’|_{-1,\gamma}^{2}+|u_{III}^{-}|_{-1,\gamma}^{2})+\delta\gamma|u_{II}’|_{-zr}^{2}1,\}
|2\gamma f\langle\Lambda^{-1}\cdot k_{II1}u_{I}^{-\prime}, u_{II}\rangle_{0,\gamma}|\leq C\gamma((\delta\gamma)^{-1}|u_{I}^{-}|_{0,r}^{2}+\delta
\leq C
(
\delta^{-1}|uI-|_{0}^{2}
,
\gamma+\delta
r
r
|uII’|_{-1}^{2}
,
.)
\gamma
|uII’|_{-\not\equiv,\gamma}^{2}
),
|2\gamma f\langle\Lambda^{-1}\cdot k_{IIIII}, u_{III}^{-}, u_{II}’\rangle_{0.\gamma}|\leq Cr((\delta \mathcal{T})^{-1}|u_{III}^{-}|_{0,\gamma}^{2}+\delta \mathcal{T}|u_{II}’|_{-1,\gamma}^{2})
\leq C
(
\delta^{-1}|uI-II|_{0}^{2} \gamma+\delta\gamma |uII’|_{-p}^{2}2
,
’
\gamma
).
The other terms are estimated by the same way as above. Thus we obtain
for large
\gamma
{\rm Im}
\langle R_{0}u_{II}, u_{II}\rangle_{0,\gamma}
\geq-C(d_{2}, Q)((\delta\gamma)^{-1}|g|_{\S,r}^{2}+\delta^{-1}|u_{I}^{-}|_{0,r}^{2}+(\delta r)^{-1}|u_{III}^{-}|_{\xi,\gamma}^{2}+\delta T|u_{II}’|_{-i^{f}}^{2},)
-|k_{II1}^{*}u_{II}’|_{0,\gamma}^{2}-2d_{1}{\rm Re}
-2|Qu_{II}’|_{0,f}^{2}+{\rm Re}
\langle Qu\text{\’{I}}_{I}, u_{II}’\rangle_{0,\gamma}
\backslash /(d_{2}Q^{*}Q) u_{II}’
+2\gamma f{\rm Im}^{/}(\backslash Q\Lambda^{-1})u_{II}’
-C(d_{2},f, Q)|u\prime II|_{-\frac{1}{2}}^{2}
,
,
u_{II/0,\gamma}^{\prime\backslash }
u_{II/0,\gamma}^{\prime\backslash }+{\rm Re}(\epsilon b)u_{II}’
,
u_{II/0,\gamma}^{\prime\backslash }
,r’
where the last term of the right arises from the commutators.
, using the above inquality,
From (8. 3) and the estimates
and
we see that
(P_{I}^{-})
(P_{III}^{-})
||P_{1} U||_{0,\gamma}^{2}+|g|_{\frac{21}{z},\gamma}\geq CT^{2}||U||_{0,r}^{2}
+C_{1}\gamma \{-|k_{III}u_{II}|_{0,\gamma}^{2}’-|k_{III}^{*}u_{II}|_{0,\gamma}^{2}’-2|Qu_{II}’|_{0,\gamma}^{2}
-2d_{1}{\rm Re}
\langle Qu\text{\’{I}}_{I}, u_{II}’\rangle_{0,\gamma}+
{\rm Re}
\backslash /(d_{2}Q^{*}Q)
u_{\acute{I}I}
,
u_{II/0,\gamma}^{\prime\backslash }
(8. 5)
+
=C2
{\rm Re}
\backslash /(\epsilon b)
|| U||_{0}^{2}
,
u_{II}’
,
u_{II/0,\gamma}^{\prime\backslash }+2
\gamma+C_{1}\gamma\{-|k_{I}
II
rf
{\rm Im}/(Q\Lambda^{-1})u_{II}’
u_{II}’|_{0,\gamma}^{2}-|k_{i_{1}}Iu’II|_{0,f}^{2}
-2|Qu_{II}’|_{0,r}^{2}+\langle R_{1}u_{II}’, u_{II}’\rangle_{0,\gamma}\}
,
,
u_{II/0,r}^{\prime\backslash }\}
T. Ohkubo and T. Shirota
144
with
in Lemma 7. 3.
From Lemma 6. 7 we have for
R_{1}\in S_{+}^{0}
(x’, \tau’, \sigma’)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap\overline{\Sigma_{-}})
k_{1II}(x’, \tau’\sigma’)=\mu’k_{1II}^{(1)}(x’, \mu’\sigma’)+k_{III}^{(0)}(x’, \sigma’)
k_{II1}(x’, \tau’\sigma’)=\mu’k_{III}^{(1)}(x’, \mu’\sigma’)+
where (6. 5) is valid for
k_{III}^{(0)}
and
k_{III}^{(0)}
|\mu’|^{2}=
k_{I11}^{0)}(x’, \sigma’)
. Furthermore
(\epsilon’)^{2}+
(\gamma’)^{2}
;
,
note that
,
and that the Q satisfies the conditions (7. 5), (7. 6) and (7. 7) in Theorem
7. 1 by virture of Lemmas 6. 4 (i), 6. 1 ) (i) and 6. 5. Then by the same
method as in the proof of Lemma 7. 4, changing (7. 4. 2) slightly one can
is sufficiently
(in this case
must be chosen so that
choose
large) and
so that the bracket of the second
term of the right hand of (8. 5):
\beta
d_{1}e_{21}^{(0)}(x_{0}, \sigma_{0})
d_{1}
d_{1}
d_{2}(x, \tau’, \sigma’)=d_{2}^{(0)}(x’, \sigma’)+
\{
\epsilon’ d_{2}^{\iota 1)}
\}\geq-C|u_{II}’|_{1}^{2}-_{2},\gamma-C_{\backslash }\prime\prime/\gamma_{0}(D’)u_{II}’,
u_{II/0,\gamma}’\backslash
, where C and C’
with
for every
are some constants corresponding to those in Lemma 7. 4.
Thus by the same way as in the proof of Theorem 7. 1 we obtain for
u_{II}=\psi(D’)v_{II}\in H_{1,\gamma}(R_{+}^{n+1})
v_{II}\in H_{1,\gamma}(R_{+}^{n+1})
U={}^{t}(^{t}u_{I}^{+t},u_{I}^{-}, \psi(D’) v_{II}, tuIII+, tu_{III}^{-})\in H_{1} \gamma(Rn+1+)
,
||P_{1} U||_{0,r}^{2}+|g|_{1}^{2},r\geq C_{1}\mathcal{T}^{2}||U||_{0,\gamma}^{2}2
for large \gamma>0 , where
is a positive constant depending only on the support of , which implies (3. 4). Thus Theorem 1. 1 is proved for k=O .
to the equations
8. 3. To obtain (1. 1) for k\geq 1, apply the operator
. We obtain
,
and
of (P^{0}, B) and put
C_{1}
\psi
\Lambda^{k}
u_{k}=\Lambda^{k}u
g_{k}=\Lambda^{k}g
f_{k}=\Lambda^{k}f
D_{n}u_{k}=Au_{k}+T_{1}\Lambda^{-k}u_{k}+f_{k}
,
Bu_{k}=T_{2}\Lambda^{-k}u_{k}(x’, 0)+g_{k}
,
(8. 6)
where by virture of Lemma 2. 1
||
|
\alpha
) (iii)
T_{1}A^{-k} u_{k}||_{0,r}\leq C ||uk||_{0,r}
T_{2}\Lambda^{-k}u_{k}|_{z}1,r\leq C|u_{k}|_{-1}
2’ r
.
Hence from (1. 1) for k=O we obtain
C\gamma||u_{k}||_{0,\gamma}\leq||u_{k}||_{0,\gamma}+|uk|_{-\not\in} \gamma+ ||f_{k}||_{0}
,
, \gamma+|gk|1 , r
.
On the other hand it follows from the same method as in the proof of
Lemma 7. 2 that
|uk|_{-z}1,\leq C\gamma
(
||u_{k}||_{0,r}+\gamma^{-1}||f_{k}||_{0}
,r
).
On
structul\wedge es
of certain
Thus we obtain for large
L^{2}
145
-well-posed mixed problems for hyperbolic systems
\gamma
C\gamma||\Lambda^{k}u||_{0,r}\leq||\Lambda^{k}f||_{0,\gamma}+|\Lambda^{k}g|_{\frac{1}{2},\gamma}
.
Use this for k=1. Then we have
\mathcal{T}||D_{n}u||_{0,\gamma}\leq r(||Au||_{0,\gamma}+||f||_{0,\gamma})
\leq r(C||\Lambda u||_{0,\gamma}+||f||_{0,\gamma})
\leq c(||f||_{1,\gamma}+|g|_{\frac{3}{2},r})
.
From this the theorem follows for k=1 :
cr||u||_{1,\gamma}\leq||f||_{1,\gamma}+|g|_{s}\tau’ f^{1}
To obtain (1. 1) for k=2 , we differentiate (8. 6) with respect to
and
use the fact that D_{n}A (x, D’ ) =AD_{n}+A_{1} , where
is the pseud0-differential
operator with symbol
. Thus (1. 1) holds for any integer k\geq 0
and the proof of Theorem 1. 1 is completed.
x_{n}
A_{1}
(D_{n}A)(x, \tau, \sigma)
\S 9. Adjoint problems.
Let b_{i}’(x’)={}^{t}(b_{i,1}’, \cdots, b_{i,2m}’)(x’)(i= 1, \cdots, m) be a certain real base of the
space of null B(x’) whose elements are
and constant outside some
compact set of R^{n} (here we may assume the existance of such a base.) Set
C^{\infty}(R^{n})
T(x’)=(b_{1}, \cdots, b_{m}, b_{1}’, \cdots, b_{m}’)
where
b_{i}(x’)={}^{t}(b_{i.1}, \cdots, b_{i,2m})
and
B^{*}(x’)=(b_{1}, \cdots, b_{m})
B(x’)T=(B(b_{1}, \cdots, b_{m}) ,
Since rank B=m ,
\det BB^{*}\neq O
O,
\cdots
,
E_{1}
is the
m\cross m
. Then it holds that
O ) =(BB^{*}, O)
.
. Hence we have
(BB^{*})^{-1}BT=(E_{1}, O)
where
,
.
identity matrix. Therefore the problem
(P^{0}, B)
in
R_{+}^{n+1}
is
equivalent to
\tilde{P}^{0}\tilde{u}=(ED_{n}-\sum_{j=0}^{n-1}(T^{-1}A_{j}T)D_{j})\overline{u}=T^{-1}f
,
(\tilde{P}^{0}, E_{1})\{
(E_{1}, O)\tilde{u}=(BB^{*})^{-1}g
where \overline{?1}(x)=T^{-1}(x’)u(x) . Let
have the Green’s formula:
(9. 1)
for
\tilde{p}\#
on
R^{n}
,
be the formal adjoint of
\tilde{P}^{0}
. Then we
(\tilde{P}^{0}\tilde{u},\tilde{v})_{L^{2}(R^{n+1})}-(\tilde{\iota x},\tilde{P}^{\#}\tilde{v})_{L^{}(R^{n+1})}=i\langle?l,\tilde{\tau r}\rangle++L^{2}(R^{n})
\tilde{u},\hat{v}\in C_{0}^{\infty}(\overline{R_{+}^{n+1}})
. Let us consider the
adjoint boundary value problem:
T. Ohkubo and T. Shirota
146
in
on
\tilde{P}’\tilde{v}=\overline{\phi}
(\tilde{P}’, E_{2})\{
(O, E_{2})v\sim=\psi
where
\tilde{P}’
is the principal part of
R_{+}^{n+1}
R^{n}
\tilde{p}\#
,
,
and
E_{2}=E_{1}
.
Then this is equivalent
to
P’(x, D)v=(ED_{n}- \sum_{-,j-,0}^{n-1}A_{j}^{*}(x)D_{j})v
(P’, B’)\{
)
in
=(ED_{n}-tA (x, D’) v=\phi
on
B’(x’)v=\psi
R^{n}
R_{+}^{n+1}
,
,
where we put
v=(T^{*})^{-1}\tilde{v}
(9. 2)
,
\phi=(T^{*})^{-1}\tilde{\phi}r
’
B’(x’)= ( O , E2)
T^{*}=(b_{1}’, \cdots, b_{m}’)^{*}
.
Now we have the following
THEOREM 9. 1. Under the conditions (I), (II) and (III), the following
,
:
a priori estimate holds with some constants
C_{k}
(9. 3)
||P’ v||_{k,-\gamma}+|B’ v|_{k+\frac{1}{2}}
\gamma_{k}>0
f\geq C_{k}\gamma ||v||_{k,-\gamma}
,-
and integer k\geq O .
PROOF of THEOREM 9. 1. Putting x_{0}=-x_{0} in
following problem :
for every
\gamma\geq\gamma_{k}
,
v\in H_{k,-r}(R_{+}^{n+1})
P^{(*)}(x, D)v=\phi
(P^{(*)}, B^{(*)})\{
B^{(*)}(x’)v(x’, 0)=\psi
(P’, B’)
in
on
we consider the
R_{+}^{n+1}
R^{n}
,
,
where we put
P^{(*)}(x_{0}, x’, x_{n} ; D_{0}, D’, D_{n})=P’(-x_{0}, x’, x_{n} ; -D_{0}, D’, D_{n})
(9. 4)
=
\{
B^{(*)}(x_{0}, x’)=B’(-x_{0}, x’)
EDn-tA (-x_{0}, x’, x_{n} ; -D_{0}, D’) ,
.
From (9. 4) and the proof of Theorem 1. 1 it suffices to show that each
of the assumptions (I), (II) and (III) for (P^{0}, B) implies the same for (P^{(*)} ,
B^{(*)}) respectively.
Note that the characteristic equation is
\det P^{(*)}(x_{0}, x’, x_{n} ; \tau, \sigma, \lambda)
(9. 5)
)
)
=
\det
=
\det (E\lambda-A(-x_{0}, x’, x_{n} ; -\tau, \sigma)
=
\det (E\lambda-A_{0}(-x_{0}, x’, x_{n})(-\eta+i\gamma)
(E\lambda-{}^{t}A(-x_{0}, x’, x_{n} ; -\tau, \sigma)
- \sum_{f=1}^{n-1}A_{f}(-x_{0}, x’, x_{n})\sigma_{j}
=O .
)
On structures of certain
-well-posed mixed problems for hyperbolic systems
L^{2}
147
Then it is easy to show that the condition (I) is fulfilled for (P^{(*)}, B^{(*)}) .
Since (P^{0}, B) ,
are so, the condition
are equivalent and (P’, B’) ,
(P^{0},
B)
(III) for
together with (9. 1) and (9. 4) implies the same for (P^{(*)} ,
(Theorem 2 of [8].) Therefore we have only to show that the condition (II) is fulfilled for (P^{(*)}, B^{(*)}) .
First we seek the Lopatinskii determinant and coupling coefficients for
(P^{(*)}, B^{(*)}) .
Put
(\tilde{P}^{0}, E_{1})
(\tilde{P}’, E_{2})
B^{(*)})
S_{0}(x, \tau, \sigma)=(h_{I}^{+}, h_{II}^{+}, h_{III}^{+}, h_{I}^{-}, h_{II}’, h_{III}^{-}),\cdot
(9. 6)
(B_{+}, B_{2})(x’, \tau, \sigma)=B(x’)S_{0}(x’, O, \tau, \sigma)
is a 2 m\cross 2m matrix and
where
holds that
S_{0}
B_{+}
\lambda II+
1
,
B_{2}
are
. ..... ....
\ldots
,
matrices. Then it
m\cross m
\alpha
M_{III}^{+}..O\lambda_{I}^{-}
(S_{0}^{-1}AS_{0})(x, \tau, \sigma)=
\backslash \cdot\lambda_{I}^{+}0
\lambda_{II}^{-}..\cdot.\cdot..\cdot.M_{III}^{-]}0
,
(b_{ij})(x’, \tau, \sigma)=(B_{+})^{-1}B_{2}
Hence we have
(S_{0}^{*}A^{*}(S_{0}^{*})^{-1})(x, \tau, \sigma)=(\begin{array}{llllll}\lambda_{I}^{+} 0 \overline{A_{II}^{\neg+}} \vdots (M_{III}^{+})^{*} \vdots O \overline{\lambda_{I}^{-}} 0 \overline{\alpha} \cdots \cdots \overline{\lambda_{II}^{-}} (M_{III}^{-})^{*}\end{array}\}-
Put
(9. 7)
(B_{2}’, B_{+}’)(x’, \tau, \sigma)=B’(x’)(S_{0}^{*})^{-1}(x’, O, \tau, \sigma)
,
are m\cross m matrices. Then we see that the Lopatinskii
where
determinant and coupling coefficients of (P^{(*)}, B^{(*)}) are
B_{+}’
B_{2}’
R^{(*)} (x_{0}, x’, \tau, \sigma)=\det B_{+}’(-x_{0}, x’, -\overline{\tau}, \sigma)
,
(9. 8)
(b_{ij}^{(*)})(x_{0}, x’, \tau, \sigma)=((B_{+}’)^{-1}B_{2}’)(-x_{0}, x’, -\overline{\tau}, \sigma)
respectively. Now we show the following two lemmas.
LEMMA 9. 1. For the quantities defifined above the following hold:
T. Ohkubo and T. Shirota
148
(i)
\overline{R^{(*)}(x_{0},x’,\tau,\sigma)}
.
\det B_{2}(-x_{0}, x’, -\overline{\tau}, \sigma)
=-R( xO, x’, -\overline{\tau}, \sigma)
(ii)
\cdot\det\overline{B_{2}’(-x_{0},x’,-\overline{\tau},\sigma)}
(b_{i}.(\begin{array}{l}*j\end{array}) )(x_{0}, x’, \tau, \sigma)=-\overline{(b_{j_{\sigma}^{}})(-x_{0},x’,-\overline{\tau},\sigma)}
,
.
ProoF. From (9. 2) we have
B(B’)^{*}=Ot
Hence it follows from (9. 6) and (9. 7) that
O=(BS_{0}S_{0}^{-1}(B’)^{*})(-x_{0}, x’, O, -\overline{\tau}, \sigma)
=
(B_{+}, B_{2}) (B’(S_{0}^{*})^{-1})^{*}=(B_{+}, B_{2})
=B_{+}(B_{2}’)^{*}+B_{2}(B_{+}’)^{*}
(\begin{array}{l}(B_{2}’)^{*}(B_{+}’)^{*}\end{array})
.
This implies (i). Furthermore we have
(b_{\dot{\iota}f})(-x_{0}, x’, -\overline{\tau}, \sigma)=(B_{+})^{-1}B_{2}(-x_{0}, x’, -\overline{\tau}, \sigma)
=-((B_{+}’)^{-1}
. )
B_{2}’
(-x_{0}, x’, -\overline{\tau}, \sigma)
.
=-(b_{ij}^{(*)}.)^{*}(x_{0}, x’, \tau, \sigma)
This shows (ii) and completes the proof of the lemma.
and
LEMMA9.2. The zeroes of
.
coincide in
PROOF. Let us consider the following constant coefficients problems
:
with parameters
R^{(*)} (x_{0}, x’, \tau, \sigma)
R(-x_{0}, x’, -\overline{\tau}, \sigma)
\Gamma\cross\overline{C_{/}}-\cross R^{n-1}
(\tau, \sigma)\in C_{-}\cross R^{n-1}
(\tilde{P}^{0}, E_{1})_{x^{0}}\{
(
ED_{n}-\tilde{A}(x^{0}, \tau, \sigma)
)
\tilde{u}=O
(E_{1},0)\tilde{u}=\tilde{g}
(ED_{n}-\overline{{}^{t}A}(x^{0}, \overline{\tau}, \sigma))v\sim=O
(\tilde{P}’, E_{2})_{x^{0}}\{
(0, E2) v\sim=\psi
in
on
in
on
x_{n}>0
,
x_{n}=O
,
x_{n}>O
,
x_{n}=O
,
\cdot
where
\tilde{A}(x, \tau, \sigma)=T^{-1}(x’)A(x, \tau, \sigma)T(x’)
,
{}^{t}\tilde{A}(x, \overline{\tau}, \sigma)=T^{*}(x’)tA(x, \overline{\tau}, \sigma)(T^{*}(x’))^{-1}
=T^{*}(x’)
A^{*}(x, \tau, \sigma)
(T^{*}(x’))^{-1}
be exponentially decaying solutions of
respectively. Then it holds that
Let
\tilde{u}
and
\tilde{v}
(\tilde{P}^{0}, E_{1})_{x^{0}}
O=((ED_{n}-\tilde{A})\tilde{u},\tilde{v})-(\tilde{?l}, (ED_{n}-\tilde{{}^{t}A})\tilde{v})
(9. 2. 1)
=i/\backslash u(O)
,
\tilde{v}(O)_{/}^{\backslash }=i(_{\backslash }^{/}\tilde{g},\tilde{v}_{1}(O)_{/}^{\backslash }+\tilde{u}_{2}(/O)\backslash ’
\psi
/).
and
(\tilde{P}’, E_{2})_{x^{0}}
,
On structures of certain
L^{2}
149
-well-posed mixed problems for hyperbolic systems
On the other hand we see that the fact
(9. 2. 2)
R(x^{0}, \tau^{0}, \sigma^{0})\neq 0
for some
(\tau^{0}, \sigma^{0})\in
is equivalent to that there exist a constant
such that for every
solution of
(\tau, \sigma)\in U(\tau^{0}, \sigma^{0})\cap
C_{0}
\overline{\overline{C}}_{-}\cross R^{n-1}
and a neighborhood
, and exponetially decaying
U(\tau^{0}, \sigma^{0})
(C_{-}\cross R^{n-1})
\tilde{g}
(\tilde{P}^{0}, E_{1})_{x^{0}}
\tilde{u}
|
.
u_{2}(\tau, \sigma, O)|\leq C_{0}|\overline{g}|
Hence if (9. 2. 2) is valid we have from (9. 2. 1)
|v_{1}( \overline{\tau}, \sigma, 0)|=\sup_{\tilde{g}}\frac{|_{\backslash }^{/}\tilde{g},\tilde{v}_{1}(O)_{/}^{\backslash }|}{|\tilde{g}|}
– \sup_{\tilde{g}}\frac{|_{\backslash }^{/}\tilde{u}_{2}(0),\psi_{/}^{\backslash }|}{|\tilde{g}|}\leq C_{0}|\psi|
,
and exponentially decaying
is
, where
. This implies
or (P’, B’)_{x^{0}} for
determinant of
,
is arbitrary we see that R(x’, \tau, \sigma)\neq 0
,
. The converse is also valid by the same discussion
view of (9. 4) and
for every
\tau\in U(\tau^{0}, \sigma^{0})\cap(C_{-}\cross R^{n})
(\tilde{P}’, E_{2})_{x^{0}}
(\tilde{P}’, E_{2})_{x^{0}}
\tau^{0}
\overline{\tau}
\sigma^{0})\in I
R’(x’,\overline{\tau}, \sigma)
R’(x^{0}, \tau^{0}, \sigma^{0})\neq O
of
solution
a Lopatinskii
,
Since
implies R’(x’ ,
as above. In
\tilde{v}
(x^{0}
(x’, \overline{\tau}, \sigma)\in\Gamma\cross\overline{0_{y}}\cross R^{n-1}+\cdot
\cross\overline{C}_{-}\cross R^{n-1}
\sigma)\neq 0
R’(-x_{0}, x’, -\tau, \sigma)=R^{(*)}(x_{0}, x’, \tau, \sigma)
,
we obtain the lemma.
Now we show the condition (II) for (P^{(*)}, B^{(*)}) . If the characteristic
, then in
equation (9. 5) has a double root
at
only on the
a neighborhood of the point (9. 5) has a real double root
surface
\lambda
(x_{0}, x’, x_{n} ; \tau^{0}, \sigma^{0})\in\Gamma\cross R^{n}
\lambda
(9. 9)
\tau=-\theta(-x_{0}, x’, x_{n} ; \sigma)
.
i s empty for the point
If the set II fo r
.
is so for the point
then that for
(II)
: Let
be a point for which the set II for
i s empty. From Lemma 9. 1 we have for every j, k\leq m
P^{(*)}
(x_{0}, x’, x_{n} ; \tau^{0}, \sigma^{0})\in I
P^{0}
\alpha)
\cross Rn,
(-x_{0}, x’, x_{n} ; -\tau^{0}, \sigma^{0})
(x^{0}, \tau^{0}, \sigma^{0})\in I
\cross R^{n}
P^{(*)}
R^{(*)} (x_{0}, x’, \tau, \sigma)
(9. 10)
.
\frac{k}{\det(V_{1}^{+},\cdots,V_{j}^{-},\cdots,V_{m}^{+})}(-x_{0}, x’, -\overline{\tau}, \sigma)
–
=-\overline{R(-x_{0},x’,-\overline{\tau},\sigma)}
. \det
j
(W_{1}^{+}, \cdots,\check{W}_{k}^{-}, \cdots, W_{m}^{+})(x_{0}, x’, \tau, \sigma)
where
(V_{1}^{+}, \cdots, V_{m}^{+}, V_{1}^{-}, \cdots, V_{m}^{-})(-x_{0}, x’, -\overline{\tau}, \sigma)
=(BS)
(-x_{0}, x’, -\overline{\tau}, \sigma)
,
T. Ohkubo and T. Shirota
150
(W_{1}^{-}, \cdots, W_{m}^{-}, W_{1}^{+}, \cdots, W_{m}^{+})(x_{0}, x’, \tau, \sigma)
=(B’ (S*) -1)
(-x_{0}, x’, -’\overline{\tau}, \sigma)\uparrow
Assume that for all j, k\leq m
\det
(W_{1}^{+}, \cdots,\check{W}_{k}^{-}j, \cdot.., W_{m}^{+})(x^{0}, \tau^{0}, \sigma^{0})=O
.
Note that it holds that for all j, i\leq m
\det
(W_{1}^{+}, \cdots,\check{W}_{i}^{+}j, \cdots, W_{m}^{+})(x^{0}, \tau^{0}, \sigma^{0})=0
.
Since
rank
we obtain for every
v_{i}\in R^{m}
\det (v_{1}, \cdots, v_{m})=O
.
This is a contradiction. Hence there exist indeces
\det
Since
,
(W_{1}^{+}, \cdots, W_{m}^{+}, W_{1}^{-}, \cdots, W_{m}^{-})(x^{0}, \tau^{0}, \sigma^{0})=m
j_{0}
,
k_{0}(\leq m)
such that
(W_{1}^{+}, \cdots,\check{W}_{k_{0}}^{-}j_{0}, \cdots, W_{m}^{+})(x^{0}, \tau^{0}, \sigma^{0})\neq 01
R(-x_{0}, x’, -\overline{\tau}, \sigma)
is not identically zero, (9. 10) for
(j_{0}, k_{0})
implies
k_{0}
that \det
small \gamma>O
(V_{1}^{+}, \cdots, \check{V}_{j_{0}}^{-}, \cdots, V_{m}^{+})
i s not identically ze ro . Hence w e obtain for
|R(*)(x_{0},x’,\tau^{0}-i\gamma,\sigma^{0})|\geq C|R(-x_{0},x’, -\tau^{0}-ir,\sigma^{0})|\geq Cr
with some C=C(x_{0}, x’, \tau^{0}, \sigma^{0})>0 , where the last inequality follows from
for (P^{0}, B) .
the condition (II)
(II)
: Let
be a point for which the set II for
.
is not empty. By the same method as in the proof of (II)
we have
(P^{0},
B)
for
the order relation :
from (II)
\alpha)
\beta)
(x^{0}, \tau^{0}, \sigma^{0})\in I
\cross R^{n}
P^{(*)}
\alpha)
\beta)
|R(*)(x^{0}, \tau^{0}-i\gamma, \sigma^{0})|\geq C\mathcal{T}^{1}z1
Hence it follows from Lemma 6. 1
\beta
) (i) that
\det(B^{(*)})’(x^{0}, \tau^{0}, \sigma^{0})\neq 0
.
On the other hand we see from Lemma 9. 1
b_{IIII}^{(*)}(x_{0}, x’, \tau, \sigma)=\frac{\det(B^{(*)})’}{R^{(*)}}=-(\overline{\frac{\det B’}{R}})
Since from Lemma 6. 1
\beta
) (i)
\det B’(-x^{0}, -\tau^{0}, \sigma^{0})\neq 0
,
\cdot
.
On structures
of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
151
we obtain
(9. 11)
R^{(*)}(x_{0}, x’, \tau, \sigma)=- C(x’, \tau, \sigma)
\cdot\overline{R(-x_{0},x’,-\overline{\tau},\sigma)}
,
where C(x’, \tau, \sigma)=\det (B^{(*)})’(\det B’)^{-1}\neq 0 . Hence from (9. 9) we see that
the second part of the condition (II)
are valid for (P^{(*)}, B^{(*)}) .
(II)
: We can define the reflection coefficients
for (P^{(*)}, B^{(*)})
by the same way as in (9. 8). Then we see that
\beta)
\gamma)
\tilde{b_{i}}
(b_{ij}^{\tilde{(*})})(x_{0}, x’, \tau, \sigma)=-(\overline{b_{fi}^{-})(-x_{0},x’,-\overline{\tau},\sigma)}
as in Lemma 9. 1. Hence if
is real then
is real. This implies
(II) . Thus Theorem 9. 1 is proved.
From (9. 1), Theorem 1. 1 and Theorem 9. 1 w e obtain the -wellposedness of (P, B) in half space which is described in the following form
(Theorem 1 and Remarks 1 of [8] applied to first order system.):
THEOREM 9. 2. Assume the conditions (/), (//), and (III) for hdf space
and \Gamma=R^{n} . Then for any integer k, s(k\geq O) there exist
positive constants
and
such that for any
and for any f\in
H_{k,s} ;
,
the problm (P, B) has a unique solution u\in
which satisfifies
b_{IIII}^{-}
b_{IIII}^{\overline{(}*)}
\gamma)
L^{2}
R^{1}\cross\Omega=R_{+}^{n+1}
C_{k,s}
\gamma_{k,s}
\gamma(R_{+}^{n+1})
\gamma\geq\gamma_{k,S}
g\in H_{k+s+\frac{1}{2},\gamma}(R^{n})
H_{1+k,s-1;\gamma}(R_{+}^{n+1})
||u||_{1+k}
,s-1; \gamma\leq Ck,s\gamma^{-1}
(||f||_{k.s;\gamma}+|g|_{k+s+_{2}^{1},r})
Furthermore if s\geq 0 and f=g=O for
x_{0}\leq T
then u=O
\S 10. Remarks on the conditions (I )\mbox{\boldmath $\beta$}) and
.
for
x_{0}\leq T.
(I )\mbox{\boldmath $\gamma$}).
10. 1. We remark in this subsection that the condition (II) -) is necessary for the case where (P, B) is a second order problem with real constant coefficients. This interesting fact is proved by R. Agemi using [2].
Let us consider the following second order problem of real constant
coefficients :
\beta
in
( \frac{\partial^{2}}{\partial t^{2}}-\Delta)u=(D_{n}^{2}-(D_{0}^{2}-\sum_{j=1}^{n-1}D_{f}^{2}))u(x)=f
(\square , B)
,
\{
on
Bu|_{\Gamma}=(D_{n}- \sum_{j=1}^{n-1}b_{f}D_{j}-cD_{0})u(x’, O)=g(x’)
where
R_{+}^{n+1}
D_{0}=-i \frac{\partial}{\partial t}
.
By virture of the transformation:
v_{1}(x)=\Lambda u
we have the following equivalent
and
v_{2}(x)=D_{n}u
2\cross 2
system
,
(\square , Q)
:
\Gamma=R^{n}
,
T. Ohkubo and T. Shirota
152
P(D_{n})\hat{v}=[E_{II}D_{n}-\Lambda_{r}(\begin{array}{ll}0 1-(-\tau^{2}+|\sigma|^{2})\Lambda_{\overline{r}}^{2} 0\end{array})](\begin{array}{l}\hat{v}_{1}\hat{v}_{2}\end{array})
(\coprod, Q)
=\{\begin{array}{l}0\hat{f}\end{array}\}
(x_{n})
Q\partial_{1}(O)+\hat{v}_{2}(O)=\hat{g}
where
lem
for
,
,
,
Q( \tau, \sigma)=-(j\sum_{-}^{n-1}b_{f}\sigma_{f}+c\tau)-1\Lambda_{\gamma}^{-1}
(\coprod, Q)
x_{n}>0
(x_{n})
. Lopatinskii determinant
R of the prob-
is
R(\tau, \sigma)=(Q, 1
)
\cdot{}^{t}(\Lambda_{\gamma}, \lambda_{II}^{+})
(10. 1)
=( \lambda_{II}^{+}-(\sum_{j=1}^{n- 1}
b
.
\Lambda_{\overline{r}}^{1}
j\sigma j+c\tau
))
\Lambda_{\overline{r}}^{1}
.
where
(\sqrt{1}=-1.)
\lambda_{11}^{\pm}(\tau, \sigma)=\pm\sqrt{\tau^{2}-|\sigma|^{2}}
Apply
\square
to Lemma 3. 1 then we have
(the case (a)),
\mu=\mu(\tau, \sigma)=\tau-|\sigma|
. Since the problem
hence we have
(I), we have the following
\theta(\sigma)=|\sigma|
LEMMA 10. 1.
\mu(\tau^{0}, \sigma^{0})=\tau^{0}-|\sigma^{0}|=O
felfills the condition
Assume the condition {III). Let
. Then it holds that
and
R(\tau^{0}, \sigma^{0})=O
where
\sigma^{0}\neq 0
(10. 2)
ProoF.
(\coprod, B)
c^{2}= \sum_{f=1}^{n-1}b_{j}^{2}
.
Since we have from (10. 1)
(10. 1. 1)
R(\tau^{0}, \sigma^{0})=R(|\sigma^{0}|,
\sigma^{0})
=-( \sum_{f=1}^{n-1}bj\sigma^{0}f+c
|\sigma^{0}|
)
\Lambda_{\overline{r}}^{1}=O
,
we see from Corollary 6. 2 and (6. 7. 2) that
grad R (
(10. 1. 2)
\theta(\sigma^{0})
,
\sigma^{0})=
grad
R(|\sigma^{0}|,
\sigma^{0}
)
=-((b_{1}+c \frac{\sigma_{1}^{0}}{|\sigma^{0}|}),
\cdots
,
(b_{n-1}+c \frac{\sigma_{n-1}^{0}}{|\sigma^{0}|}))\Lambda_{f}^{-1}
=O .
Hence if c=O we have b_{j}=O for all j=1,
from (10. 1. 1) and (10. 1. 2) that
c^{2}= \sum_{j=1}^{n1}b_{f}^{2}
.
\cdots
, n –1.
If
c\neq O
, it
follows
On structures
of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
153
This proves the lemma.
Next we show that the condition (II)
is necessary for the problem
(\coprod, Q) to be
-well-posed:
THEOREM 10. 1. Under the same assumptions as in Lmma 10. 1, we
have
\beta)
L^{2}
co\dim\{(x’, \sigma)\in R^{n}\cross R^{n-1}
=
rank
;
R(\theta(\sigma),
\sigma^{0}
He,ss(x,\sigma)R(\theta(\sigma^{0}),
\sigma)=O\}
)
for c\neq 0 ,
for c=O .
\int^{=n-2} ,
1=0,
ProoF. From (10. 1. 2) we have
\frac{\partial^{2}R}{\partial\sigma_{j}\partial\sigma_{k}}
(
\theta(\sigma)
,
\sigma)=-c\frac{\partial}{\partial\sigma_{f}}(\frac{\sigma_{k}}{|\sigma|})\Lambda_{f}^{1}
c \sigma_{k}(\frac{+\sigma_{j}}{|\sigma|^{3}})\Lambda_{\gamma}^{-1}
for
j\neq k
for
j=k .
,
=\{
c( \frac{-1}{|\sigma|}+\frac{\sigma_{j}\sigma_{k}}{|\sigma|^{3}})
\Lambda_{\overline{r}}^{1}
Hence
He_{Gss}R(\theta(\sigma),
= \frac{+c}{|\sigma|^{3}}
\sigma
)
(\begin{array}{lllll} \sigma_{1}-|\sigma| \sigma_{1}\sigma_{2} \sigma_{1}\sigma_{n-1} \vdots \sigma_{2}^{2}-|a|^{2}.
\sigma_{2}\cdot\sigma_{1} t \sigma \vdots\sigma \sigma^{2} -|\sigma|^{2}\end{array})
\Lambda_{\overline{r}}^{1}
n-l
If c=0
rank
If
c\neq 0
He” ss(x\sigma)(\theta(\sigma),
\sigma)=O
.
this matrix has the eigenvalues
\underline{-c|\sigma|^{-1},\cdots}\underline{-c|\sigma|^{-1}},
,
O
(see
n-2
Hence
rank
Put
He” ss(x\sigma)R(\theta(\sigma^{0}),
\sigma^{0})=n-2
.
\S 4 in
[2].)
T. Ohkuho and T. Shirota
154
M_{R}=\{(x’, \sigma)\in R^{n}\cross R^{n-1}
;
=R^{n}\cross\{\sigma\in R^{n-1}
we have from Lemma 10. 1
that
It follows from
If^{1}c=0
;
R(\theta(\sigma),
\sigma)=O\}
\sum_{j=1}^{n-1}b_{j}\sigma_{j}+c|\sigma|=O\}\uparrow
b_{i}=O
and hence
M_{R}=R^{2n-1}
. Let
c\neq 0
.
R(\theta(\sigma), \sigma)=0
c^{2}| \sigma|^{2}=(\sum_{f=1}^{n-1}b_{j}\sigma_{j})^{2}
Hence from (10. 2) we have
| \sigma|^{2}(\sum_{j=1}^{n-1}b_{j}^{2})=(\sum_{j=1}^{n-1}b_{j}\sigma_{j})^{2},\cdot
i.e .
|\sigma|
where
(10. 1. 1) that
b={}^{t}(b_{1}, \cdots, b_{n-1})
.
Lec
.
|b|=|\langle\sigma, b\rangle
c>0
M_{R}=R^{n}\cross\{\sigma\in R^{n-1}
;
for
|\sigma|
.
|
,
simp1icit^{\acute{\iota}}y
|b|=|\langle\sigma, b\rangle
|
,
,
then it follows from
\sum b_{f}\sigma_{f}<0\}
.
Hence \dim M_{R}=n+1 (also, in the case c<0 ) for fixed b . Therefor we
obtain co\dim M_{R}=n-2=rankHess\sigma R(\theta(\sigma^{0}), \sigma^{0}) , for c\neq 0 .
Thus the proof is completed.
10. 2. This subsection is devoted to the generalization of our Main
theorem to the case where the boundary operator B(x’) is complex. Since
the condition (II)
has an essential meaning in the case where B is real,
and ) described in \S 1 instead of (II)
we consider the conditions (II)
and ). Throughout this subsection, we assume the condition (I) with
complex valued B and assume all symbols are homogeneous of order 0.
and (6. 4. 3) it follows that for
From (II)
, x’\in U\{X)
\gamma)
\gamma’
\beta’)
\beta)
\gamma
(\tau, \sigma)\in\overline{\Sigma_{-\cap}}U(\tau^{0}, \sigma^{0})
\beta’)
(10. 3)
R_{II}(x’, \tau, \sigma)=(r_{1}(x’, \mu, \sigma)+\sqrt\overline{\mu}r_{2}(x’, \mu, \sigma)
)
(\sqrt{\mu}-D(x’, \sigma)
),
where
and
are smooth and
. Furthermore let Q be the
function defined in Lemma 6. 1 ) (i) . Then it follows from (6. 4. 3) and
(6. 1) that
is real if Q is real and
is real or pure imaginary in the
case (a) or (b), respectively. Applying implicit function theorem to (10. 3)
we see from Lemma 6. 4 (i) that (II)
and ) imply that |D is real or
pure imaginary in the case (a) or (b), respectively. This shows that (II)
is fulfilled. Since it follows from (6. 4. 3) that (II)
and ) imply (II)
, we see that (II)
implies (II)
and ).
Hereafter we consider
as
.
r_{1}
r_{1}(x^{0}, O, \sigma^{0})\neq 0
r_{2}
\beta
R_{I1}
\sqrt{\mu}
\beta)
\gamma
\beta’)
\gamma’)
\beta)
\beta)
\beta’)
(\tau, \sigma)
\gamma’
vectors\in\overline{\Sigma_{-}}
\beta’
On structures
of certia n
L^{2}
-well-posed mixed problems for hyperbolic systems
155
LEMMA 10. 2. Using the notations in Lemma 3. 1 and (7.1) Q is written in the form : except terms with respect to
\gamma
Q(x’, \mu, \sigma)=(2^{-1}(1+e_{12}\epsilon)^{-1}(e_{11}-e_{22})+r_{2})\epsilon
in the case (a),
-((1+e_{12}\epsilon)^{-1}\lambda_{2}+r_{2}D)D
\{
or
-(-i(1+e_{12}\epsilon)^{-1}\lambda_{2}’+r_{2}D)
D
in the case (b)
respectively.
Proof. It follows from (6. 1), (6. 4. 3) and (10. 3) that
Q=(r_{1}+\sqrt{\mu}r_{2})(\sqrt{\mu}-D)-(\mu s_{1}^{(1)}+\sqrt{\mu}s_{2})
s
=-r_{1}D+\mu r_{2}-\mu 1(1)+ (r_{1}-
Since Q is analytic in
\mu
\sqrt{\mu}
.
we have
r_{1}-r_{2}D-s_{2}=0
Since
r_{2}D-s_{2})
,
Q=-r_{1}D+\mu r_{2}-\mu s_{1}^{(1)}
=(r_{2}-s_{1}^{(1)})\mu-(r_{2}D+s_{2})
D.
This together with (3. 1), (3. 3), (6. 1) and (7. 1) implies the lemma.
Put Q^{(0)}=Q(x’, O, \sigma) and
7. 4. Then we see from Lemma 10. 2 that
Q^{(1)}= \frac{\partial Q}{\partial\mu}(x’, O, \sigma)
=-(\lambda_{2}^{(0)} (x’, \sigma)+r_{2}^{(0)}(x’., \sigma)
Q^{(0)}\{
as in the proof of Lemma
D(x’, \sigma)
) Dx’ ,
or
=-(-i(\lambda_{2}’)^{(0)} (x’, \sigma)+r_{2}^{(0)}(x’, \sigma) D(x’, \sigma)
(10. 4)
\sigma)
)
D(x’, \sigma)
,
Q^{(1)}=(2^{-1}(e_{11}^{(0)}-e_{22}^{(0)})+r_{2}^{(0)})(x’, \sigma)
e_{12}^{(0)}-\lambda_{2}^{(1)}-r_{2}^{(1)}D)(x’, \sigma)\cdot D(x’, \sigma)
\int_{or}^{+(\lambda_{2}^{(0)}}
[
+(-i(\lambda_{2}’)^{(0)}e_{12}^{(0)}+i(\lambda_{2}’)^{(1)}-r_{2}^{(1)}D)(x’, \sigma)\cdot D(x’, \sigma)
, in the case (a) and (6)
with the same kind of notations as
and
respectively. Now Theorem 7. 1 is generalized in the following
and (III). Thm the
THEOREM 10. 2. Assume the conditions (II) ,
Q^{(0)}
Q^{(1)}
\alpha)
\beta’)
conclusion of Theorem 7. 1 is valid.
PROOF. We have only to show that Lemma 7. 4 is valid under the
conditions of this theorem. We see from the proof of Lemma 7. 4 that
T. Ohhibo and T. Shirota
156
it
Let us
to show that the expression (7. 4. 7) is non-negative.
in the right of (7. 4. 7):
the coefficient of
suffiffiffices_{arrow}
consider
d_{1}
k_{4}
\{-2k_{3}{\rm Re} Q^{(0)}+({\rm Im} Q^{(0)})^{2}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)}))
(10. 5)
+ ({\rm Re} Q^{(0)})^{2}(2{\rm Re} Q^{(1)}-(e_{11}^{(0)}-e_{22}^{(0)}))\}k_{3}+O(|Q^{(0)}|^{3})
Use (10. 4), the relation
case (a) to
=k_{3}
+2
k_{3}=e_{21}^{(0)}+2{\rm Re}\overline{Q^{(1)}}Q^{(0)}
.
, then the above equals in the
\{2(\lambda_{2}^{(0)}{\rm Re} D+{\rm Re} r_{2}^{\prime 0)}D^{2})(e_{21}^{(0)}+2{\rm Re}\overline{Q^{(1)}}Q^{(0)})
(\lambda_{2}^{(0)}{\rm Im} D+{\rm Im} r_{2}^{(0)}D^{2})^{2}{\rm Re}(r_{2}^{(0)}+D(\lambda_{2}^{(0)}e_{12}^{(0)}-\lambda_{2}^{(1)}-r_{2}^{(1)}D))\}
+0 (({\rm Re} D)^{2})+O(|D|^{3}) .
Since
e_{21}^{(0}
)
=(\lambda_{2}^{(0)})^{2}
e_{21}^{(0)}
it holds that
{\rm Re} r_{2}^{(0)}D^{2}+ (\lambda_{2}^{(0)} {\rm Im} D+{\rm Im} r_{2}^{(0)}D^{2})^{2}{\rm Re} r_{2}^{(0)}
=
(e_{21}^{(0)} {\rm Re} r_{2}^{(0})
{\rm Re} D-2e_{21}^{(0)}{\rm Im} r_{2}^{(0)}
{\rm Im} D){\rm Re} D+O(|D|^{3})
.
Hence (10. 5) equals to
=2k_{3}^{2}\lambda_{2}^{(0)}
{\rm Re} D+O(|D|){\rm Re} D+O(|D|^{3})
.
In the case (b), we see by the same way that (10. 5) equals to
=2k_{3}^{2}(\lambda_{2}’)^{(0)}{\rm Im} D+O(|D|){\rm Im} D+O(|D|^{3})
.
Therefore it is non-negative (or non-positive in the case (b)) in a sufficiently
. Moreover from
, because of the condition (II)
small
U(x^{0})\cross U(\sigma^{0})
\beta’)
(10. 4) we have |Q^{0}|^{4}=O(|D|^{4}) . Consequently, we see that there exist real
such that (7. 4. 7) is
satisfying (7. 1. 1) and a neighborhood
U(x^{0})\cross U(\sigma^{0})
d_{1}
non-negative. Thus the theorem is proved.
Using the above theorem let us generalize Theorem 1. 1:
and {III). Then
THEOREM 10. 3. Assume the conditions (II) , ,
the conclusion of Theorem 1. 1 is valid for complex valued B(x’) .
, it follows from the condition
and
PROOF. Since
is equivalent to \mu=O and D(x’, \sigma)
(II)
that R(x’, \mu, \sigma)=O for
=O . Hence we see from the proof of Lemma 6. 6 that Lemma 6. 6 is
that \eta=\theta(II) and
valid for such points
D(x’, \sigma)=O .
On the other hand, we see from (1. 2) that in some
in Lemma 10.2
of a point
\alpha)
{\rm Im}\sqrt{\mu}\geq 0
\beta’)
\gamma’)
{\rm Re}\sqrt{\mu}\leq 0
{\rm Im}\mu\leq O
\beta’)
(x’, \eta, \sigma)\in U(x^{0})\cross(U(\tau^{0}, \sigma^{0})\cap R^{n})
\sigma)
U(x^{0})\cross
U(\sigma^{0})
(x^{0}, \sigma^{0})
{\rm Re} D(x’, \sigma)\geq 0
or
-{\rm Im} D(x’, \sigma)\geq 0
On structures of certain
L^{2}
-well-posed mixed problems for hyperbolic systems
157
in the case (a) and (b) respectively.
Hence we obtain
gr,ad(x,\sigma){\rm Re} D(x’, \sigma)=O
or
gr” ad(x\sigma){\rm Im} D(x’, \sigma)=O
whenever
{\rm Re} D(x’, \sigma)=O
or
{\rm Im} D(x’, \sigma)=O
respectively. Consequently it is seen from the condition (II)
inequalities (6. 5) are replaced by
|k_{II}^{0)}(x’, \sigma)|2
,
\mathcal{T}’)
that the
|k_{III}^{0)}(x’, \sigma)|^{2}\leq C{\rm Re} D(x’, \sigma)
or
(10. 6)
|k_{II}^{0)}(x’, \sigma)|^{2}
,
|k_{III}^{0}
(x’, \sigma)|^{2}\leq-C{\rm Im} D(x’, \sigma)
in the case (a) and (b) respectively.
In the proof of (1. 1) in subsection 8. 2, use these inequalities instead
of (6. 5). Then we see from the proofs of Theorems 7. 1 and 10. 2 that
(1. 1) is valid for our case.
Thus we can generalize our Main theorem:
and (III). Then
THEOREM 10. 4. Assume the conditions (II) , ,
the conclusion of Main theorem is valid for complex valued B(x’) .
PROOF. Follow the proof of Theorem 9. 1 and note the relation (9. 11)
and ) are preserved
especially. Then it is seen that the conditions (II)
for the problem (P^{(*)}, B^{(*)}) . Therefore the assertions of Theorems 9. 1 and
9. 2 are valid under the conditions of this theorem.
\alpha)
\beta’)
\beta’)
\mathcal{T}’)
\gamma’
Department of Mathematics
S\={o}ka University.
Department of Mathematics
Hokkaido University.
References
[1]
R. AGEMI: On energy inequalities of mixed problems for hyperbolic equations
of second order, Jour. Fac. Sci. Hokkaido Univ., Ser. I, Vol. 21, 221-236
(1971).
[2]
R. AGEMI: Iterated mixed problems for d’Alembertian, Hokkaido Math. J. Vol.
3* No. 1, 104-128 (1974)
T, Ohkubo and T. Shirota
158
[3]
[4]
[5]
[6]
[7]
[8]
[9]
R. AGEMI and T. SHIROTA: On necessary and sufficient conditions for L^{2_{-}}we11posedness of mixed problems for hyperbolic equations, Jour. Fac. Sci.
Hokkaido Univ., Ser. I, Vol. 21, 133-151 (1970).
R. AGEMI and T. SHIROTA: On necessary and sufficient conditions for
posedness of mixed problems for hyperbolic equations II, ibid, Vol. 22,
137-149 (1972).
M. S. AGRANOVICH: Boundary value problems for systems with a parameter,
Math. USSR Sbornik, Vol. 13, 25-64 (1971).
L.
RMANDER: Linear pratial differential operators, Springer-Verlag (1963).
H. O. KREISS: Initial-boundary value problems for hyperbolic systems, Comm.
Pure Appl. Math., Vol. 23, 277-298 (1970).
K. KUBOTA: Remarks on boundary value problems for hyperbolic equations,
Hokkaido Math. J. Vol. 2, 202-213 (1973).
P. D. LAX and L. NIRENBERG: On stability for difference schemes, A sharp
rding inequality, Comm. Pure Appl. Math. Vol. 19, 473-492
form of
L^{2_{-}}we11\llcorner
H\dot{o}
G[mathring]_{a}
(1966).
-well posedness for hyperbolic mixed problems, Publ. R.I.M.S.,
Vol. 8, 265-293 (1972).
[111 T. SHIROTA: On the propagation speed of hyperbolic operator with mixed
boundary conditions, Jour, Fac. Sci., Hokkaido Univ., Ser. I, Vol. 22, 2531 (1972).
[12] T. SHIROTA: On certain L^{2_{-}}we11 -posed mixed problems for hyperbolic system
of first order, Proc. Japan Acad., Vol. 50, No. 2, 143-147 (1974).
[10]
R. SAKAMOTO:
L^{2}
(Received May 29, 1974
S
’